It seems like the value of a board member is their role with the company, not their control over the company. Those seem like two distinct concepts, and you only want to deal with the former.
Why can't founders give people equity stakes on par with what a "board member" would receive, while giving them no voting power within the company? Is this a case of "In theory that would work, but in practice you wouldn't get the benefits an actual board member would bring"? If so, why is that?
Maybe there are few people who would actually bring enough value to your company where it makes sense to offer them a board seat. If there are few, then they can demand a board seat. And maybe that's why voting power goes along with their positive effect: It's part of the deal. But I'm just wildly guessing, and it'd be great to hear from someone with insights about the actual reasons one can't simply give up an equity stake and expect to receive help on par with an actual board member.
In the Fortune 500, it's a prestige gig as much as anything. For many smaller companies, agreeing to be on the board while not being a large investor, is almost akin to being involved in a local govt board of economic development for a town or county - it's sometimes done as a favor, out of quasi-charity to help out, and or you know some of the people involved and want to see the company succeed.
It can also be a great way to make more connections in business; one board membership refers to the next - you can kind of move up an economic chain that way if you go on a successful ride-along. People like to be associated with / involved with successful enterprises.
The historical reason for connected board control, is at least two fold: enabling trusted parties to oust the CEO without needing to hold a large vote by shareholders (not an easy thing to do now, and more difficult a century ago); and enabling direct major shareholder control without requiring a wider shareholder vote (if you own 10,000 shares of Walmart along with a million other investors, while technically those total shares have the same voting value net as the hedge fund that owns 5% of the company - the 5% entity wants a lot of influence due to the concentrated stake). Board control also enables rapid action in cases of fraud or abuse by the top executive/s, in which case time may be critical - that is, you don't want to wait months for a full shareholder vote.
Also, who do you leave legal authority to, when it comes to deciding the replacement for a CEO that has decided to retire or quit (or dies, or whatever)? Certainly shareholders as a whole have legal control, but it's not very practical to have thousands of shareholders trying to figure out who gets to be the next CEO (most of those shareholders will have absolutely no idea who would make a good candidate, or where to start, having typically zero direct involvement in understand the business top to bottom, and most shareholders will have no experience running a serious business). The only entity that makes sense in that regard is a board of some sort.
They get paid $100-250k for maybe 1 day a month of work and very little responsibility most of the time. That's a lot of money in absolute terms but maybe not relative to the total income of typical board members.
There is also no constraint on 'founderness' vs 'boardness' which is to say, if you're a small company with a 3 person board and you hire a CTO or a CMO or some other C level position you can put them on the board as well.
A similar concept can be used in the case of one founder + multiple investors. The founder holds the legal right to appoint numerous board seats (eg 3), and the two investors get a right to appoint one seat each. In a case of coup or other problems - and assuming everything is done properly legally - the founder appoints the remaining seats to their loyal faction and removes any of the other problematic board members. That's not to say things won't be messy regardless in a bad version of that scenario, in which one or more investors on the board tries to oust the founder.
E.g. 2 founders + 1 investor is very common for Series A stage startups right now. Those with one founder tend to have two votes for the founder and one vote for the investor.