(Sorry for the wall of text!)
Yes, printing more money for hospitals would liquefy the whole economy into jello. We can't solve our problems by printing more money. Inflation is very bad, in particular it's bad for the working class who don't have the luxury of having non-monetary assets.
"Why fund this and not that"
This is the very thing I raised as an argument against TPOASIWID. If a hospital isn't saving enough cancer patients, it's because it doesn't have enough money. Now, money doesn't explain everything of course -- perhaps there are some racially-linked diseases that are underfunded, and here "TPOASIWID" serves an explanation, but a rather bad one because it doesn't actually explain anything. A better explanation might be "it's due to institutional racism," or "it's simply an oversight," or "the technology to cure this disease doesn't exist yet" -- and then you might look at why it doesn't exist and you'll be looking at something other than the hospital. "The purpose of the hospital is to have institutional racism" doesn't make sense at all -- it's the purpose of institutional racism to infect the hospital, and the purpose of leftists to purge institutional racism.
"why do we collect taxes instead of letting inflation do the exact same thing?"
This presents the best case for TPOASIWID in my opinion. That's an interesting question too. (I suppose the answer is that taxes benefit the working class of course, whereas inflation benefits the wealthy, who hardly have any money when compared to their investments.) But I don't see how saying "The purpose of taxes is what they do" leads one to querying about inflation. I'm not an expert about the economy, but taxes seem to me like they more or less do the thing they're supposed to be doing so TPOASIWID does seem to match here, at least when compared to "why not just use inflation for taxation." (Neither of these methods will touch tax-averse autocrats of course.)
Does it match because TPOASIWID is good at prediction? No -- it simply discounts the possibility that a system could be failing, so of course it looks accurate when a system is apparently not failing. It's a brazen assertion that whoever is running the world is doing things exactly right and simply can't fail. This seems insane to me because things are hard. Take the USA's "War on Drugs" -- in my view and probably that of most people, it was a spectacular failure in retrospect. What was the purpose of the war on drugs? Well to answer that we should look at who wanted it to happen and why. In my opinion, it was spurred on by many people who just really wanted to eradicate drugs, and didn't have ulterior motives. But TPOASIWID just leads one to conspiracy theories: since the War on Drugs basically just got a lot of black people thrown in jail, then surely all those people who claim to hate drugs must actually just hate black people, and not drugs at all! After all, if they actually hated drugs, then WoD would have been successful.
But I'll admit that if you don't know the purpose of a system, figuring out what it does is a good place to start.