I am an owner and very passionate about the breed.
Here are the two I have had. They are fantastic horses: as brave as anything and kind gentle and loyal. BUT not if you are pushy and try to treat them like a TB. They need firm, fair handling and they will be putty in your hands. They take a while to trust a new person and will sometime simply not like some people for no obvious reason.
They can be a bit bargy - I think they forget how big they are - but are fine if reminded to behave. Harry, (Mosscarr Rhinehart) my gelding, had some Parelli training and will happily walk at my side on a loose rope and isn't ever much trouble. Hanna, the mare, was trained for the showring from a yearling so was good to handle also, but she knew her strength and would chest her way through stable doors and gates!
They are excellent hunters and all rounders, they do well at dressage (Spring Pascal is CBx), being surprisingly light on their feet, and unsurprisingly, are excellent endurance horses as well as driving and agricultural work.
They can carry a lot of weight and are as solid as a rock: unspooky and sensible. They are intellegent horses, so when they don't understand something, they'll stop and try to work it out, which sometimes leads to them being described as stubborn: not something I have ever found. Trying to explain to seventeen hands of horse can be challenging, but if they don't understand what you're asking, find a way they do understand and they are very happy to oblige.
They are indeed hardy: Harry is out in a moorside field at the moment without a rug and will stay out all winter, only being rugged when it gets very rough. They are usually good doers and have feet hard enough to not need shoes. Having had Cleveland Bays, I'd never be satisfied with anything else (except perhaps Fell ponies) which seems to be the case with most owners!
They are getting extremely rare: there are less than 300 breeding mares left WORLDWIDE which is a very sad state of affairs. There's a couple of groups on Facebook that might be worth looking at: Cleveland Bay Wales (not limited to Wales) and Cleveland Bay Horses Buy or Sell. I'm sure there are others, but I know both of those are good.