I have a mental check list that I have gone down before I even go look at them, if they meet all those criteria I will go and meet them and then its all down to the vibe (I've mostly brought mine as youngsters) if I get a good vibe I buy, if I don't I walk away. Occasionally the vibe isn't too strong one way or the other and you spend time with them and ride/do groundwork for half hour to see how they learn, 9/10 times I end up walking away from those ones though.
I think often people look for a type or look and let that love at first sight thing over-ride the practical side, those criteria we all set when we start looking which is more about the horse we need rather than want.
When helping a friend look she wanted a cob, def. no TB's or arab's as that's what everyone tells you when you buy your first, 8-10 years & 15-15.2hh or something like that, been there done it type that she, as a nervy advanced novice, could have fun with hacking. We looked at dozens of horses and her shortlist was 2 cobs;
The 1st was a 15.2 very handsome skewbald HW cob with lots of feather and mane, 7 year old, novice ride, who from a trekking center that was closing.
The 2nd was a 14.3, 9 year old, palomino, stocky cob type, been there got the T-shirt, who's owner had to sell due to young children.
So both meet her criteria and were handsome, and she was ready to part with her money but I had to tell her no on both counts once we saw/rode them, horse 1 had been used as a very novice ride, he hadn't learnt to canter yet
and he panicked when alone and the steering and brakes failed, nothing nasty but so green it was untrue, he was also so obese that he had a 2in deep gully down his entire back. Horse 2 I guessed at twice the age advertised, probably had PPID/EMS from the puffy eyes and odd fat pads, was an absolute tank in a gag and grackle, again not nasty but a confident teenagers pony. She ended up with a OTTB, who within minutes of meeting and 30 seconds of sitting on I knew she would love because he was just so chilled out, he failed as a racer as he just couldn't be bothered to run a lot but was nice and intelligent and learnt well