We observe that politicians are happy to cut taxes (for people who can benefit them) if they personally get paid as much or more than before. Why would it be otherwise?

Having the ability to take and redistribute someone else's money provides a concentrated benefit to the one doing the taking and redistributing. Cutting taxes produces a much more diffuse benefit. Concentrated benefits lead to Machiavellian behavior much more than diffuse benefits. It is possible, of course, to have an anti-taxes lobbying group which provides a concentrated benefit, but the overall balance between concentrated and diffuse benefits is on the side of the higher taxes.

(And any long-term interest, eg power for their family, should take the state of their civilization into account.)

That would be a diffuse cost. The politician may care about the portion of the diffuse effectthat affects his family, but that's only a small portion of the total. If the politician makes policy based on which costs help him and his family and which ones hurt him and his family, the concentrated ones will win. The ones that affect all civilization, a small portion of which he actually cares about because it goes to his family, will lose.