installing a mod mid playthrough breaks your saves.
No it doesn't. Not if done right.
just because the save loads without crashing doesn't mean it didn't get corrupted.
If you are lucky then the corruption is somewhere where you never notice. but those can add up over time.
Also, how does one "do it right"?
only way I know is to know exactly what files each mod changes (as well as save file modifications) and about all possible incompatibilities and issues. Which is humanly impossible task
There is a reason all modders strongly recommend you not make any changes once you started a playthrough