Emotionally, it sucks to take a hit and not own anything, but I don't want to fall into the sunk cost fallacy, and it would be nice to be out of the landlord game on a house that doesn't cash-flow positive.

Business does not mix and the outcome can be emotionally devastating.

One more tip – people remember stories that trigger their emotions.

As much as my home comes with the emotional attachment of all the work I have done, I do my best to look at it as a tangible asset and how I can best utilize the equity for growth.

Avoid the cash offer stuff, and saying quick closing in letters, use emotion, people really do respond well to it.

Is this an emotional and preference thing or a true need?

We have simple form letter that says "the information you provided does not meet the requirements for an Emotional Support Animal".FWIW, you could also change your no-pet policy.  

We keep it professional and try not to get emotional about it.

Also this emotional pressure is really common with high pressure sales environments, which is often the case with large wholesaling organizations. 

+95% of agents only have experience dealing with owner-occupied transactions - which are mostly EMOTIONAL decision based.You can ID these types of agents easily by asking them:1) If they are RE investors themselves.