My answer may be a bit different than others, for better or worse. I definitely agree with the sentiment of "go with your gut, stay safe, and don't feel bad saying no."
I often accept last minute stay requests or accept dogs without a meet and greet. I've even accepted dogs after some pretty intense meet and greets. For me, frankly, it depends on my mood when I get the request, what I have going on during the stay timeframe, and how much time I have the day of the dropoff to dedicate to good introductions. If I'm in a spontaneous mood, don't have a lot of stressful meetings/obligations/work going on, and am available for a good chunk of time when they want to drop off the dog, then for me it's a go.
I wouldn't recommend this approach for everybody, maybe not even most sitters. I am quite confident in my abilities to manage difficult dogs, read body language, manage difficult behaviors that arise (barking, etc), facilitate good doggie meetings, and overall have a good stay. I've had dogs that did not get along in meet and greets that ended up playing with my dogs throughout the stay (after doing walks together and setting clear boundaries with intensely close supervision for safety reasons). This may be more work than you're willing to put in, and risk, which is completely fine.
Also as a side note, last-minute stays do not automatically mean the dog is a problem, although there is definitely that risk. (Although to be fair, I've had some great meet and greets that turned into very stressful stays.) In my experience, these owners had other plans for dog care that fell through or had an emergency come up and needed care for their dog. However, it is definitely worth asking about in a phone call or message prior to the stay about what led to the last-minute search. I would be more concerned about an owner that didn't give a reason for the last-minute search for care.
That's my two cents. Hope it helps! Good luck!