....is it a terrible idea to rent a scooter over there?
Hard to be objective with a question like this, but I suppose some experience would help you along.
If you drive the automatic scooters that are 110-125ccs (Honda Scoopy, Yamaha Fino/Filano, Honda Moove, Honda click, Yamaha Nouvo, etc....) within the last 10 model years or so^, and
- you drive at a speed that is compatible with road conditions
- stay to the left when in doubt
- keep your head on a swivel
You should be fine. Practice on local side streets first before going into crowded areas. In fact, I would recommend FIRST using a motorbike taxi a few times, then ride with a local friend 2ND, then ride by yourself.
@Gnusper has a point in that Westerners tend to be neurotic. But you will see Thai-style driving is NOT the same because the thinking and safety considerations for self and others are structurally different. #1 reason I see tourist who get into motorbike accidents here are because they try to drive like a Westerner in Thailand.
It's unlikely that you will get travel insurance that will properly cover motorcycle driving in Thailand. Don't be surprised if in your search the insurance contract specifically excludes Thailand or has ridiculous restrictions on riding motorcycles. If you do find it, please leave it in the comments or update your answer.
I hope I didn't scare you. Motorbike is a [relatively] cheap and convenient way to move around quickly. You will have lots of fun and most tourist will never have a serious incident or notice anything too out of the ordinary during their month or so stay. Stay long enough (start growing some roots like having children) you will begin to lose some of your innocence.
^If you are over 125kg, you could opt for 150cc...anything over that for an inexperienced rider is likely overkill.