I bought an outdated condo, planning to do a cosmetic rehab later when I had more budget.I moved out of the condo in mid-2024 for a job, and by September, my tenant noticed mice droppings.

In Mid-west, 90% of deals are not true off-market.

Here in mid-west flyover country, rents are still creeping up and almost all the development has been luxury rentals (mostly by Redwood).

Would it be a bad idea to buy a single family home built in mid 1930's?

If they are leaving mid-Feb and willing to pay 6 weeks' rent to break the lease, that is reasonable. 

I started in mid February.  

The property is located in a mid-size city with several attractions nearby.I understand that management fees can vary by location, but this property management company charges 20%.

Mid-term rentals is also a strategy worth exploring, a good book to read on that is "30 day stay" by Zeona McIntyre & Sarah Weaver.  

Also see if monthly gap might be lower as a mid- or short-term rental.

when you get to the rust belt and mid west no real need for it in a big way when you can still buy nice SFRs for 150 to 300k all day long.