I’m not a carpenter or structural engineer, so I can’t guarantee my advice is proper, but I have a couple thoughts - first that those brackets would probably never pass inspection. In fact, I don’t think the stairs as built would either. Anyway, my gut is telling me the treads should have full support on both sides front to back, so you could find some matching millwork and glue & screw bracing underneath the treads on both sides. I still don’t think that would be ideal, and takes away from that floating MCM look. Another idea would be to contact an iron monger to create full depth brackets you could have installed more invisibly by anchoring them between the stringer and tread. May I also suggest taking the opportunity to install continuous handrails for better safety and code compliance?

I found some illustrations to help. The photo of the split staircase has too much exposure. There must be a way to make them more discreet, but you can at least see what I mean. Another more radical choice would be to replace the stringers with iron or steel that is engineered with built in support, but that’s a budget-buster for most and probably isn’t realistic. Here are those photos.

These stairs break all the rules and are not structurally safe. I cannot believe that they went unnoted by even a hack "home inspector".

I have to agree. The more I look at your photos, the more your stairs really give me anxiety. My strongest advice is to look into replacing them entirely. There is nothing safe about the way they’re built.

Please get a few experienced carpenters to examine them to see if they are up to code; then you can make your decision about what to do.

Where you located? If the IRC2018 building code is adopted in your area you cannot have more than 4" of open risers.

"To fix them I was thinking to add long screws on the sides from the outside of the stairs in to each step to make them stronger since some of the steps are wobbly. Does this seem to be a good idea ?"

No, because it would not only be ugly, you'd be screwing into the end grain of the treads, one of the weakest possible connections. Speaking of weak connections, the small brackets appear to be installed with drywall screws which are very weak in shear. If you're going to reinforce these, I'd suggest the blocking (wood or metal) between the tread and stringer as pictured previously; no end grain attachment and doesn't look bad.

I wouldn't add any expensive replacement handrail or guardrail to these cheap stairs. Do 'em over comprehensively or patch and forget about it please.

Those need professional help to do right and those brackets are just all wrong.

To fix them I was thinking to add long screws on the sides from the outside of the stairs in to each step to make them stronger since some of the steps are wobbly. Does this seem to be a good idea ?

Nope. All the more so because there is a gap between the end of the tread and the connection to the stringer. All of the stress will be on that short section of screw that isn't into the meat of the wood. And screws don't get a good bite in end grain.

Under different circumstances, where the two pieces are snugged up tight to one another, this is one way to fasten into end grain.

https://woodgears.ca/shop-tricks/endgrain_screw.html

But fastening into end grain is a general no-no and best avoided in the design phase.