I've got a Strat outfitted with a Mahler, a high end Ibanez with a Rose, & a Strat with the old Fender design vibrato.
I prefer the locking design, but I still use that other Strat. (The first has an ash body with a Birdseye maple neck & fingerboard & Duncan high gain pickups. The second has an alder body, but rosewood fingerboard & Vintage Noiseless P/Us, so they sound very different)

I have no such tuning issues with any of them.

I mostly want the whammy, but I've got a Tele, SG copy (very old Ibanez), Dan Armstrong, & a Schecter C-1 Blackjack for fixed bridge.

The only vibrato I don't like is the one on my Jazzmaster, but I barely ever play that guitar. Too noisy.

Some thoughts:
1. The bridge saddles on your guitar: are any at the end of the throw? That matters a lot.
2. Adding springs don't help if the guitar is slightly out of intonation. It actually doesn't help at all, except that it makes it harder to move the whammy block, so there's less slack & stress.
3. If you really don't want to use the whammy, to get really stable, get a piece of wood, and whittle it to a wedge that can be put behind the knife-edge. This prevents the block from moving at all. The more stable, the better it stays in tune. Ask Les Paul players! (I'm just not a Gibson guy. I like the longer scale, but they are fine guitars)
4. Back to 2, I'm leaning toward an intonation issue. If a Floyd Rose gave everyone the problem you find, people would have quit buying them 40 years ago. Who wants that constant hassle?

If you have the gear, get the intonation correct. If you don't, there are trustworthy guitar techs everywhere. Have them do it.