I will tell you this honestly because I bought the wrong horse. I had been ridding since young and could only afford a horse at when I was 22.My local riding school who knew me well sold her to me, which goes to show that everyone is out to get you when te want to off load a bad horse. She started bad, e.g. bucking and taking off, and finished off unhandlable and unriddable,. It took me 4.5 years to accept that I had to give up on her and the whole process dented my confidence enormously.
Do not look for another livery yard, do not ride this horse. Write, recorded delivery, to the seller saying the horse is unsuitable for purpose and that she has 7 days to contact you to arrange for the horse to be returned and you to be refunded. You may have to pay a transporter to return the horse but the seller should refund you in cash on return of the horse - don't let her fob you off on this, you have the right to return the horse for a full refund.
A five year failed racehorse is not the right choice for a child that wants to have a bit of fun hacking. It is nowhere near the right choice, it won't become the right choice with a bit of work and perseverance, and anyone who knows anything about horses would tell you this (I don't mean this to make you feel bad about your choice but to point out that there is no question whatsoever that the seller saw you coming).
Next time go to your DC's riding school and ask the instructor to help you find a suitable horse, you will need to pay her a fee for her time but it will be well worth it. Ideally you want a family horse that has been doing the job you want it to do but has been sadly outgrown. You should get the horse vetted including bloods.
The horse marked is severely depressed, it is the middle of the winter, you offer a good home, you should be able to find what you want for 1-2k, especially as you don't need a weight carrier (which I assume you do not if a TB was OK weight wise before). Type, breed, colour are not as important as temperament, health and suitability for the job, but realistically the horse should be at least 8 years old to have the experience your child needs to learn from (and anything up to 14yo should be perfectly fine for a few years of fun before it is outgrown).