floor joists splitting at non-beam end
None of what you're describing is particularly concerning, and is fairly typical of old, dry wood. Those cracks have probably been there almost as long as the joists have. The brackets you propose will not provide any useful benefit, as any movement would be due to compression of the joist, and brackets won't stop that. You'd need to sister the joists fully or in part, bonding the two members together adequately with construction adhesive and screws. I don't think that's necessary, though.

However, the water damage as indicated by the staining on multiple members raises flags. Softening of wood as it decays results in settling and structural damage. That should be your primary concern. If the cause hasn't been resolved, do some investigating. My guess is that there's an entry above that area with inadequate drainage.