Definitely have a look. I bought one Mine had raced from age 4 to 16 I searched his history and it was amazing, he'd been ridden by Richard dunwoody at one point. He was my horse of a lifetime, despite having been bar fired. 110% in traffic, you could do anything with him without holding him (clipping, shoeing, injecting) and I happily plonked a child on him. Jumped like a stag (ex chaser) He was PTS age 22 with cushings and I miss him everyday
Well - there's no such thing as a cheap thoroughbred (speaking from bitter experience). They can cost more to feed, can have crap feet and can be heavy on the vet. Having said that I love them to bits and would def go and see. A plus side is that they generally box, shoe, clip and behave well in the stable having seen so much.
On a more cautious note my ex pointer reared with me out hunting - he wanted to go one way with the hounds and I wanted to go home - right up, we almost hit the tip over point and it absolutely terrified me. I've ridden for more than 40 years, since I was very small and had experience on all sorts of horses, including ones that have reared, but that was the closest I've come to go over and it scared me. I have been somewhat wary of him since and it's not an experience I would like to repeat - I now turn him in small circles if I think he's losing patience which stops him, but it's not much fun. If you are happy to deal with it then that's fine - but it is dangerous.....
I think the 'cheap' is a red herring. There are loads of cheap TBs and quite a few cheap other breeds at the moment as people are desperate to get rid of horses they can no longer afford.
'Cheap' is also misleading in the case of a horse that rears. If you do take on a horse with such an issue the first thing you would have to do is get the vet out to exclude a physical cause to his problem. You may uncover nothing, in which case you still have a potentially very dangerous behavioural issue, or you may find a physical cause some of which are very debilitating and require lengthy and costly treatments.
The question for me would not be 'shall I buy this horse?' but 'why this horse and not some other one?'. How many horses have you viewed? What do you want to do with your horse? Why does this seem particularly well suited to your purpose in comparison with others out there?
Sounds like you are not in a hurry which is a great position to be in. On the selling side anyone who can wait will be waiting for spring anyway so it's best to give it another month or two for the weather to improve.