First of all I let me say that I love Caprice. I too was nervous about her when previewed but having lived with her for over a year now she's been a constructive force in Netrunner both in deckbuilding and playing.
Here's why I love her: You're absolutely right that no amount of money can defeat her, and that's the point of Caprice. She solves the econ problem, which is that in an asymmetric game, one side will always be more efficient at generating credits. Credits are the second-most important resource in terms of being able to play the game (clicks being the first, but they are much more difficult to generate/lose.) Rezzing ice, advancing agendas/ambushes (and then triggering the ambushes!), even playing most operations all take credits. If the runner is more efficient at generating credits then a smart runner will always be richer than the corp, and this effectively kills glacier and tag'n'bag decks. Early successful corp decks in Netrunner were Fast Advance (which didn't need to have more credits than the corp, just enough to score from hand) or Never Advance (which cut the cost on bluffing, by never advancing remotes). To be fair there were other reasons why these decks saw most competitive play, but I believe this to be a major factor.
Normally the solution against glacier decks is to not run, stockpile massive amounts of credits, pack an infiltration or two, and only run when the corp tries to score an agenda. Cards like Ash, Red Herrings, Corporate Troubleshooter, etc all relied on credits to stop the steal, so if the runner plays clever with Deus X, Inside Job, Femme, Stimhack etc they can greatly reduce the cost of infrequent runs. In a similar vein efficiently costed ice doesn't help if the corp can't force the runner to run frequently. Now along comes Caprice. Suddenly no matter how cheap the runner makes the run, or how many credits they have, they're not guaranteed the steal. So now the runner has to run a server multiple times to ferret out the Caprice before the corp goes for the score. Suddenly that efficient ice is relevant again and more windows open up for the corp. And glacier decks are viable again, because (like Fast Advance) the corp doesn't need to have more credits, just enough credits to play the psi without crippling themselves.
Designwise this opens up the econ space. Instead of being very conservative with cards, trying to keep runner and corp econ across seven different factions within a small margin of each other (a ludicrously difficult task) FFG now has much more flexible tolerances for what is too good at credit generation.
Okay so maybe Caprice is good for the game, but that doesn't mean that she's fun to play against. Most people complain about her adding an unnecessary element of randomness to the game, which detracts from competitive decision making. I say this is also false. You mention that the psi-game is not a solvable problem, when actually it has been solved. Here on BGG coffeeyay uses game theory to solve the Caprice game. You should absolutely read the whole thing, but I'll try to summarize. How much the corp or runner should bid depends on the expected value of the runner winning the psi-game. The closer to infinite the value is (i.e. winning the game) the more the corp should bid evenly amongst the three options (i.e. roll a die / random) but at lower levels (e.g. Caprice protecting herself) the corp should bid 2 much more often, and 0 much less often. As a simple example, take a Caprice on HQ and the runner Account Siphons. coffeeyay breaks down what the corp should bid based on how many credits they have left in this table here. TL:DR - there is a line of bidding that is not bidding evenly between 0, 1, and 2 that will advantage the player following it (even if they're not winning the psi-games).
This ideal Caprice/psi-game strategy deepens the games playing strategy in two ways: Deviation and conditioning. Deviation in knowing when to abandon the game theory optimal play, and play for short term gains. And conditioning in establishing patterns of play to later exploit them. These are common elements of games like Yomi which is absolutely competitive.
If after all this you still hate Caprice, then still there are several runner-side work arounds. Vamp is a popular one, the corp can't play psi if they're broke. Quest Completed also eliminates the need for psi-games. Still these solutions require multiple runs (runs needed to clear Crisiums or other Caprices out of centrals) and a fair chunk of credits. Which is good because they don't upset the economic balancing effect Caprice brings to the game, and add to the strategy Caprice brings, rather than silver bullet her.
*(Sorry if this is too much, I've had this article in me for a little while and this post was a great opportunity to write it out.)