I've always thought similarly to you and the only points I can think of (and NOT say you are wrong in ANY way)
1) I typically use historical starts and the minors are bloated, I agree - however my "National Holiday" is just before the draft, thinning out the herd is literally no different by using either date as far as the lowest three leagues are concerned. I also keep all the teams unlike you - but it doesn't cost anything at that point to my $$$.
2) My "way" of thinning the herd might be a tad quicker and I offer it only as a alternative start. Intl League has any player 18 or older is either pushed up to the lowest level minor team (DSL/ASL) or released. Then for your Dominican and Arizona leagues - sort your team by age, ANYONE over the age of 22 or older probably needs to go for the same reason as you give - the clear majority aren't gonna be MLB quality and most players have no stats at that level and age - cannon fodder at best!
3) I add one year to every level starting at A (23) as my "cut line" - I especially abhor middling older players at AA/A while those youngsters sit at lower levels. I only caution that there is every so often that overage player is blocked at the lower level, then I usually will start nosing around my "shop player" and see what prospect I can get for him.
4) AAA is the magical soup pot of baseball, you can find a little bit of everything if you look hard enough
I usually use age as my barometer for AAA - can I trade a 30 yr old reserve for a 25 yr old with some upside? Can I swap my pitching needs with an "extra" position player?
5) I'm also a fan of trying to get bottom feeding teams to do 2-1 or 3-1 deals to ease my congestion at a position. I get rid of players that aren't going to push out my starter/backup and by giving 2-3 players and getting one of their 2.5 or 3 star prospects I open my 40 man roster spots and $$$ by having a 21-25 y/o prospect. Most deals are roundly criticized by the AI Asst. GM, but I find they usually work out for my plans.