Giving this power a democratic mandate and legitimacy may make the role "very different", but that doesn't mean the constitutional dynamic would be worse.
I feel it is self-evident that the new dynamic created by an elected GG would be worse. If you have a person at the head of government whose only job is to say no to things and resolve constitutional disputes they are left with only those tools to differentiate themselves from other candidates.
That creates an incentive system for this person to veto and manufacture constitutional strife in order to create a profile for themselves. It is a role built specifically for political arsonists.
On the other hand, I think such a belief requires one to hold quite a low view of the public and democratic processes generally. You might say that experience supports such a low view, but I would counter that it applies equally, or more so, to the status quo, where all power is held by the prime minister, who has every incentive (and much greater ability) to create these crises (especially just before elections) and tell the country that they have to unite behind the government to survive.
The fact that such a strategy is already possible under the current system just confirms to me that there needs to be different roles, held by different people, with different incentives, to stop these extreme edge cases. I think that what is self-evident is that concentration of power leads to abuse, while diffusion of power leads to moderation and representativeness.
I look at the American political system and imagine a version of it where the president was the president but did not in fact operate the executive, that person's only job was the power of the veto - it seems like a nightmare.
The idea that a PM has some incentive to manufacture a crisis is true I suppose, but that is a very high-risk strategy. Real-world experience indicates that popular support is a product of good economic conditions and the rare exceptions to these instances (i.e. GWB post-9/11) stand out because they are very extraordinary. (And usually end very badly!) Every political body at every level has the power to manufacture a crisis and benefit from it, but that isn't very effective because the electorate understands that they have the power to address the crisis, or prevent it in the first place so the existence of that dynamic doesn't prove much of anything from my perspective.
Personally I don't see why it would be a nightmare if the US president didn't operate the executive. I think voters would be a lot less worried about the "wrong" person getting elected if the president couldn't declare war or appoint SCOTUS judges, so this change might reduce some of the tribalism.
As you say, every political body has the power to create crises, and the electorate should quickly see through any deliberate attempts to create one, so there seems little risk that the minimal-president role I am suggesting would attract people motivated to cause trouble.