If you do not remove the programs properly in some way , they will be less fully removed. (programs do not always fully remove to begin with).
If you deleted programs in the "program files" location, the items in the registry, the appdata , the user appdata , any drivers, and possible "installer" data will not be removed. Plus other locations depending.
It would not stop the operating system from working, it would do little damage to programs that were not effected (still working). Mostly it would just leave little messes all over the computer, the same little messes that can slow the computer down a tiny bit, with them installed, and take some storage space.
Very Important "sub-systems" of the operating system that are no longer fully installed could effect the operation of programs that rely on those sub-systems.
the Removal of "Common Files" can remove enough items from some programs that those programs would no longer operate.
In ADD/Remove , many programs removed this way would not fully un-install via add/remove after pulling thier files out of Program Files.
Add/remove removing the Add/remove Entry there, because the "installer is not found", will not clean up the parts and pieces leftover.
Possible Ways to attempt to "Fix":
Put the files back, or put the files back in from a backup.
Re-install the programs , some of them will think they are still installed, which can take a lot more effort to re-install them by cleaning up something first.
If the intent was to Un-Install them, re-installing them and then Un-installing them properly, would mostly remove them correctally.
Manually finish the un-install: Try and clean up the registry and all the places where the programs parts and pieces hang out, to complete the un-installation. For some programs this would take minutes, for some it would take hours.
Give up on the whole thing, and take this opertunity to re-install the whole operating system and programs from scratch.
Leave the mess, and re-install the programs you actually need, worry about re-installing next time you re-install anyway, if you would do that anyway eventually. If specific things fail, worry about re-installing those things.
What I would do for my own system would depend on what was deleted from there.