Well it's less that the AI never saves money but mostly that they can't. Rare indeed is the realm that can stand its entire levies and men at arms at once without running short on the monthly income. The AI will try to resolve that by setting their steward to collect taxes in their most prosperous holding, but it's usually not anywhere near enough to make up the shortfall of some King draining 15 gold a month with their steward giving them an extra 0.8 a month. They'll also hire mercenaries anytime they're at war in a case where they're outnumbered which makes sense, but will further drain their coffers.
Now include the faction fix the last update did to make factions more dangerous for players who were easily handling them before that. The AI can't manage factions anywhere near as well as the average player so they will constantly end up in revolts where they have to hire mercenaries (and usually still won't matter), as well as the AI decision to always raise their full force for every single war (where a player might only raise a part as needed for flex and savings). Or the tendency to desperately go for any possible deal to get an alliance when they're hard pressed via marriage, often creating situations where foreign rulers will claim their holdings on inheritance because they didn't care over the short term reaction, and with the inability to fight off factions will never reach High/Absolute Crown Authority to prevent that...
And they got a hard uphill battle to ever have a positive cash flow as they're constantly locked in wars against factions or neighbors who are out of their own time of troubles.
All starts with the cash as I see it. The need to raise their entire army even if it's something like the French King calling up 9,000 men to service to put down an enemy of 300 levies just draining their coffers badly as they suck that loss for all the year or two they'll take on the war due to sieges or whatever. Meaning they can't afford buildings to increase their income. They can't afford to put their stewards on increasing development for long term growth. They're just desperate for raw money which they'll lose when they go to war against some peasant revolt a few months later, blowing their cash on a mercenary company because they're not sure if their 12 Knights, 1600 men at arms, and 6400 levies under a skilled commander can hope to defeat 8000 peasants.
For the regions effected, Varangian Adventures also do this because even though most of the adventures is levy trash the sheer troop count means they'll avoid battle. While still standing up their full army.
And once the backruptcy cycle starts? I never see the AI recover unless I go out of my way to basically pay off all their debts and give them enough gold to have 2000 in the bank while also upgrading their holdings (If they're my vassal).
