If you feel the home is causing you health problems, I recommend you move to another rental where you can feel safer.

The intent was to maximize small businesses' ability to maintain liquidity, retain their employees and recover from this health crisis as quickly as possible. 

I purchased under my name in cash because the agreement was time sensitive given the seller’s declining health and offer deadline.

I operate a couple recovery homes, and work with people who have mental health issues.

Since the city is involved you can try and get the health and safety department  (might be the same as building) involved and they may find the home to be in such poor shape that the tenants need to leave. 

Could I be liable for any health issues (real or perceived) that may arise with future tenants?

I think if there is any possibility of it create a health hazard (even a small one) I would address immediately. 

Now they repeatedly threatened with a lawyers demand letter inhabitable health issues for mold and other issues. 

Do one or both for the education and modify as you learn and build wealth -one thing for sure is it is not static  it changes with age, health, laws, technology, family situations, etc   Taxation, succession plan, estate plan, insurance etc may all play a role.

@Luke Petro,Try contacting the local health department inspector, buy them a cup of coffee and see their experience with exterminators.