How should we divide up expense account amounts between partners?

I'm going to answer you with my own experience. The way you mentioned to divide the expenses makes total sense and it's consider the "rational" thing to do. I have seen it work many times and it's what many would consider "fair".
The problem (and this is counterintuitive) is that we are humans with emotions and we can't separate us from them.
Once someone starts buying nicer things the "ego" hits in, also the "jealousy" and the competitive nature. This brings bad culture and a worst environment. I know you think "we are different", "it won't happen to us" but it actually does and it's not your fault, it's just our nature.

My solution is the following. Treat the company as a separate entity from the three of you. So the company (not you) have revenue and costs. THE company can have expenses and they should be as little as possible to run efficient and lean. THE company has to create the most profits as long as it's in the same direction of creating value for their clients.

Now, because the company has shareholders (you guys/gals) the profits it generates will go into your pockets 50/30/20. This is after your salaries, that depends on your place in the company and that is money totally entitled to each of you. The profits can be expended as whatever you want because it's like part of your salaries.
You will think this makes no sense due that is just a "technical" step. But it's important to separate you from your company. Keep personal and professional in each side of the table.

Hope this helped :) If you want to reach out I would be happy to talk. I have helped many family companies to also deal with this kind of issues.
Have a great day

This is really more of a philosophical/partnership question than an accounting question. At my company and those of most of my clients, if all three partners are equally active in the business and make a similar salary, then usually business related expenses are paid the same for all. If these are more “perks” (country club memberships, car payments, etc.), then a decision has to be made as to who gets what, just like what salary gets paid to each partner.

Because no one owns a majority, you are going to have to come to some sort of agreement among the partners. Even if someone owned 51% though, you would probably still want to reach a consensus. When the 51% owner makes these types of decisions unilaterally, then it tends to undermine the spirit of the partnership. So the answer is, you guys need to decide what everyone can live with and go with that. I can tell you all day that the 50% partner should get a nicer car, but that’s not going to stop the other two partners from being unhappy if they don’t like that solution. So if you are the 50% partner looking for me to give you an answer you can show to partners and say “See, everyone else thinks I should get a better car,” I’m going to advise you that’s a really bad idea all the time, not just as it relates to this topic.

The last comment I’ll make is a little bigger picture. I’d suggest not instituting a lot of perks and running a lot of semi-personal expenses through the company. Profits will be distributed at the end of the year based on your sharing ratios. If one partner wants to join a country club, they can use their profits to do that. Expenses like that aren’t deductible for tax purposes anyway, so there’s no benefit to running them through the company. Make your business about business, and leave the other stuff for people to spend on their own.