But I don't think they suit the purpose of "pet" very well, or trails for that matter. Some can be lovely but they are often very excitable and sensitive. Many of them require regular work or they can be a little difficult. They can be prone to injuries and require a lot of maintenance, like feed and rugging.
It's of course your choice, but for a pet I would look for something a bit more low maintenance. There are a lot of horses out there who need rescuing.
I have four horses and each one gets a great deal of attention and the one I can ride I ride a great deal. I am 6'4 and a plus sized rider so not every horse is for me. One of mine is in her late twenties and no longer can be ridden (and was a hot horse though I have had her since she was twelve and rode her from the boarder of Canada to the boarder of Mexico on the Pacific Crest Trail...), ones a very small arab mare and another is a friesian cross out of the arab both of which my are my husbands.
This would be a 'pet' horse to ride and ride and ride. We don't have children and are in our thirties so we have animals instead. I think this guy might be the perfect fit for me.