You say "The only things people can point to that they like about the constitution are..." and then proceed to point out things that are not even in the constitution.
Here is something to like about the constitution: a system of government that has seen the peaceful transition of power every 4 years for 244 consecutive years.
If you don't count that little incident in the 1860's
I'm sure you could get pedantic on this matter, but the subject of OP was related to structure and function of the US government. This is the purview of the constitution and not the first 10 amendments.
Functionally, they may serve a different purpose. So does the preamble, but it's still a part of the constitution. As is the bill of rights.
In any case, I think a lot of the problems you're describing are not strictly constitutional, or at least lead to contradictions if you try to fix them.
* It's very hard to keep courts from becoming politicized unless either everyone acts in good faith (ha), or you figure out a way to pick judges that doesn't ultimately get its power from elected officials. The lifetime appointment idea is about the best I can think of for that - even something like a single 20-year term has the problem of kickbacks after leaving office. For a long time, the requirement of supermajority approval was also a really good moderating idea, until McConnell realized that he could win by default by just not doing his job. Maybe a combination of large supermajorities and forced votes? But all of this runs into the problem that one party simply does not want the government to function. It's relatively easy to design a system that prevents people from doing something; it's much harder to design a system that forces them to do something when they benefit from stagnation.
* Executive power will always ebb and flow as laws are passed, unless you disallow Congress from delegating its own power, which would be silly.
* Legislature is at an impasse - I mean, if you have disagreements on the order of 40% of the population thinking that large minorities shouldn't be considered citizens, then I'm not sure how you can get past impasses. This is a symptom of hyperpartisanism, which is itself, I think, a symptom of poor education and misinformation. People disagree. You can't force them to agree unless you a priori give some of them power over others, which kind of defeats the point.
As for what's good - I think most people are also fans of the 4th and 5th amendments, and the 8th gets a reasonable number of references even in casual conversation (even if people don't know which one it is). And the 13th-15th are all pretty damn important.
The Constitutional structure as a whole is pretty good. The things I can easily think of that I /don't/ like about it, and which haven't been cleaned up in amendments, are:
1. Senate apportionment. Corn shouldn't get a vote. Admittedly, this is pretty major, but it would also be very, very hard to change in a new Constitutional convention, since even if you first agree that the Constitution should be rewritten, states won't ratify a new one if they feel like they're losing power.
2. Lack of clarity on what to do when there's a criminal in a position of power. It's tough, because any procedure can quickly become a farce if you make impeachment and removal too easy, but it would be nice if at the very least there were some minimal enforced procedures with regards to witness testimony and debate. And we could maybe strengthen the bit about the House having subpoena powers with a "yes, that means you, Donald."
3. Mandate better voting systems for Congress. As it stands, states are free to, on their own, adopt proportional representation (at least in states with sizeable delegations; Wyoming can't), ranked-choice voting, or any of the other better-than-first-past-the-post ideas that political scientists and mathematicians have come up with. This doesn't /have/ to be a Constitutional change, but it might be better if it were.
4. Mandate a formula for resizing, not just reapportioning, the House after every census. 435 is not enough to capture the variance in sizes of states without severe compression, and leaving it up to Congress to make itself more representative is a bit of an oversight.
EDIT: Also, splitting up states is wildly inefficient, to the point of being laughable as a solution to any problem. First of all, how do you choose where do you draw the line? Maybe California makes it relatively easy, since there's exactly two major population centers, and maybe you can find a way to split Texas in the desert and Florida in the swamp, but once you get to New York in the 4th cycle you're going to have a big problem. Then, you have to split up all the existing stuff the state has - who gets custody of prisoners? who gets the state treasury and/or debt? And finally, you have the simple task of setting up a second state government from scratch - just write a new constitution, have snap elections for all your officials at every level, write new building and utility codes (and figure out how to handle what you already have, especially if the new sub-state is not politically aligned with the old one), hire and train an entirely new civil service, and start collecting taxes.
There were parties in the UK by the time of the revolution. By the 1790s there were warnings being emitted about their potential toxicity in the US. But there's nothing done to formally recognize political parties or manage their risk
I always theorized the two-party system on a national level was less a function of first-past-the-post and more a function of a presidential, rather than prime-ministerial system. You need a political machine capable of running a national campaign, and the ones that are big enough to do that will also likely conquer down-ballot elections.
If you no longer have to run a national campaign, you'd end up with primarily state-level parties. The California Democrats and the Illinois Democrats would have about as much in common as the current US and German Green parties. Each state may have two main parties due to FPTP still, but they'd likely be less coordinated and financially linked, forced to form blocs and consensuses at the federal level.
In the US, we do in fact have parties that are dramatically different across state lines at the state level. This is changing a bit in the last few years, but it used to be only at the national level that the parties were internally homogenous. And even in Congress, except on the hot-button issues, there is some diversity across delegations from different states. Elizabeth Warren and Doug Jones are not the same.
The problem is, if you go to a prime minister (e.g., if Speaker of the House got the powers of the President), then the individual state elections effectively become the national election. Imagine that there are no parties, but you and your friends have a few billion dollars to throw around, and want to make sure that the Prime Speaker is Nancy Pelosi. You'll do that by supporting politicians who would vote for her (donating, but also organizing get-out-the-vote drives, etc. - campaign finance laws can't prevent this). If anyone new wants to enter politics, they'll want to take advantage of your machine, which means that if they're on the fence about a policy they'll default to agreeing with you to make sure you pick them, and oops now you've reinvented parties. It's just an inevitable consequence of two facts put together: (1) FPTP voting, and (2) campaign organization having economies of scale, both in terms of money and in terms of voter attention span.
Or do you imagine the little people like us would have a say in such a rewrite?
For the legislature, I’d go for a lower house that would be a form of proportional representation. Each representative would have voting power in proportion to how many votes they receive. There could be thresholds and limits, and could allow people to split their votes for representatives (so you could get single issue candidates in).
The upper house wouldn’t vote for bills at all. They’d be regional, and regions could easily design themselves (so a city could form its region and expand with expanding suburbs, or split off from its suburbs). They’d basically be glorified lobbyists to promote local issues, or delay a bill if their region would be adversely affected by it. Regions aren’t as important as they used to be (Phoenix and Atlanta are politically pretty similar), but you still want to preserve their voice.
This is an incredibly unrealistic idea for the same reason the Commerce Clause can be used to justify anything. How do you govern international affairs without the power to oversee implementation of treaties? How do you do anything in "other domestic policy" without affecting the economy? All of those things are intrinsically linked to each other, and trying to split them apart and consider them in isolation would be catastrophic, except for the fact that it's impossible.
I think most people consider this a very risky move.
Turkey was a thriving democracy turned slowly authoritarian by stacking the courts with political judges. Poland is in the midst of a similar transformation.