Yes, it does. Some individuals react differently. Some individuals cannot take aspirin daily for heart benefits, some cannot take ibuprofen for pain relief. Both these drugs can have reactions with other medications. It isn't an insignificant portion of the population, but we shouldn't allow those individuals create a dialogue that scares others away from trying the drug.
If anything, we just need to spend a bit more time talking to folks about the possible side effects of the medications.
It's my understanding that with any particular drug used to treat depression the proportion of sufferers who can't take it is high - certainly IME the proportions are not at all insignificant.
Aside: my understanding is that aspirin is no longer recommended (in mainstream medecine [in the UK]) to be used in a preventative way.
If anything, we just need to spend a bit more time talking to folks about the possible side effects of the medications.