How to isolated stand from springy floors?

Like the additional joist idea (sister). Use adhesive as well as nails or screws.
You could also box the joists (every other pair) with 1/2" or 3/4" plywood, again using adhesive and nails or screws.

Either of these choices should stiffen the floor considerably. The adhesive is important. Both ideas requires floor to be open underneath.

From your pic it looks like hardwood, IOW a floating floor, so how are tighting up the joists of the subfloor going to help? There will still be some "spring" in the floor.
Even with mass loading, there will still be some bounce. The floor needs to be damped, with the addition of isolation footers under the component and if possible, isolation under the shelf.

Also IME, do not put any iso feet under the rack; it is likely to affect the performance of your TT. I'll bet your TT is very happy being on a heavy stand touching the floor.

As a quick effort at low cost, what I have done is beneath those components where there is no interposed individual maple board base supporting the component, simply place three or four Cryo'ed Mini Springs between the component and the existing hard, flat base.
Even though some of your components may quiver when you walk by, there won't be a jarring shock to any of the protected components.
The sound you hear from your suspended components will surprise you with the improvement heard.
Long ago, the Aurios ball bearing approach also allowed for this quivering or rocking when footfalls affected components on my equipment stand, but the sound improvement was nowhere near the significant upgrade with the Mini Springs from Machina Dynamica.
There are variations in implementing these springs.
This minimalist approach is fast, simple, audibly impressive -- even though not the "ultimate" implementation of what is termed Baby Promethean Mini Isolators by Machina Dynamica.

The entire house is moving. So making the floor rigid will not solve the underlying problem. The ideal solution would be to isolate the house and the components.

Interestingly, the OP makes no mention of problems (skipping, feedback) with the turntable; in fact, he suggested that there is no issue with the sound. The turntable, while it has a type of suspension (rubber feet between the chassis and plinth) and is generally very good, is not known for being particularly immune to this sort of problem. If the floor were truly "springy" this would affect turntable playback. Before undertaking more drastic fixes, I would look for an extremely microphonic tube in the amplifier; replacing each tube one at a time to isolate the culprit(s) and also putting some isolation device under the amp. I would suggest that the problem is not floor springiness but a tube or an amp that does not have good immunity to low frequency excitation; something that would probably not be solved by adding mass underneath it and, in fact, could make the problem worse.