Here's what ABTA has to say:
https://abta.com/about-us/press/holidaymakers-putting-themselves-at-risk-by-hiring-quad-bikes
I have never ridden a scooter or motorbike - seen too many tourists who've removed a lot of skin when falling off. I did hire a quad in Agios Stefanos this year, but hated it and gave it back after three days. I didn't feel at all in control of it. I've owned and driven some very powerful sports cars in my time, I've driven thousands of miles over very dangerous African roads, but the quad was the only vehicle that's ever scared me.
That's the root of the concern: people who would never hire a scooter feel safe on a quad because it has four wheels. It's not as safe as one assumes, particularly (I read) when cornering or accelerating hard. Apart from the fact that the automatic gearbox was "snatchy" and prone to jerks when starting off, I found the steering remarkably un-positive and inclined to veer at the slightest pothole. You are an experienced quad-rider - I'm not, and I'm probably more typical.
I have to agree with Jimbo. Driven a quad once with my brother-in-law. Very jerky and stiff handling to such an extent that my in-law cracked a wrist trying to control the thing. He didn't crash or have an accident it was just the effort required to drive it.
I was concerned throughout about overturning as they're very heavy.
Please note this was in the UK.