A path to follow each and every time, regardless of the excuses given by the tenant paying late.  

But service dogs seem an excuse to avoid no-pet clause.

South of Boston we can get high efficiency FHW units installed for much less than the "new heating systems will sink you $10k and wipe you out" scare tactic excuse people use for not investing, not sure how that cost translates to NY.  

There are really good con artists out there who will seem so cute and so sincere, and they are really good at getting you to like them so much - that when you learn about something in their info that does not meet your criteria - you will want to overlook it or excuse it.  

Either way, it seems like answer to your problem unless you aren't charging an application fee in which case taking that money out of pocket for an excuse to reject them seems silly.

Excusing a late fee will void the lease language allowing a tenant to legally argue in the future that you may not charge late fees.

They may have "sold" it to the appliance shop to get some extra money then used that as an excuse.

Otherwise, you're opening a can of worm for many excuses you end up paying for.

One of my favorite quotes from I don't remember where is - "This is not an excuse contest."

Now if someone is moving from Michigan, that can be a legitimate excuse, but it's something to consider...One quick idea..