It needed a full renovation and was very undesirable when I purchased it.

Tuxedo Park is very undesirable.

In the desire to stick to what we want (#1) and get emotional (#3), we might find ourselves with an undesirable deal.

Is probability-adjusted cost of an increase in undesirable occupants really high enough to warrant giving up $2M in valuation on a house? 

All of these point of the neighborhood having high demand and a lack of inventory, usually because the neighborhood is in transition from undesirable to up and coming. 

The only thing peripherally I can think of is most investment loans want an LLC on title, and a lender won't want you transferring an investment residence owned by an LLC into personal ownership by you.  

One of them was run by a local investor and almost fully occupied when I took over, but after I cleared out the non-payers and general undesirables, I ended up turning over 50 of 76 units, which costs $4-5K per unit.  

If you see something undesirable, but not a deal breaker, then you will be better prepared to handle it once the acquisition is complete.4.

It gives me more confidence to be a bit more aggressive and clear out undesirable inherited tenants.

Once they have seen the property and are interested, the prequalification and background checks will do the rest of the work in weeding out undesirable tenants for you.