There's potential truth in what you're saying, but that doesn't mean that a tourism tax is
far from ideal. The economics of governments and their agencies are very different from those of companies.
Most importantly, while private companies have a market incentive to treat customers better when they pay more, governments don't. This is because governments both have a monopoly and so don't face outside competition, and because they don't face bankruptcy if they don't draw in the $200M in funds that you mentioned.
Instead, all that the federal government is saying to the TSA/other ESTA recipients is that "we made a tax which generates $200M, so we expect you to step up your services with the funding". It works only partially, because the only motivation that ESTA recipients (especially the TSA) have to improve is the word of god, and not any meaningful amounts of hard cash profit. The TSA is the worst of all with this, because unlike state marketing authorities, the TSA has no theoretical benefit from increased tourism. The smaller and more local the agencies you are analyzing get, the more like businesses they are, and so the more effective an ESTA would be.
So having an ESTA to make the TSA better is essentially just the same as increasing the TSA's funding from tax dollars and telling it to be better. Then you might as well do the latter, so as not to discourage tourists since tourists are much more likely to stop coming to the country than US citizens are to stop paying taxes because the income tax went up 0.01% to fund the $200M.
And even increasing taxes isn't ideal, because theoretically, if the TSA experience being better is worth the money, tourism will increase proportionately, and tourism's benefit to the economy will drive up tax revenues automatically without a tax.
Most ideal of all would be to find a way to privatize the TSA in a way that's safe and efficient, because then the new TSA corporation(s) would snap to market equilibrium.