Breed can be an indicator of a "general rule"... But I see it more in the sense that some breeds have purposely been bred "dumbed down" for a specific job (breeding that often require a very quiet, solid, take orders kind of horse) while others have been bred to be more "hyped up" ("hot") and others still have been bred in-between.
For instance, it would be rather impractical for a plough horse to be very highstrung - chances are, few people breed high strung drafts for that purpose. As a result, many drafts (but not all... The general rule always has exceptions!) are quiet minded and appear quite "placid". Now this does change a bit depending on which draft breed too... But for argument's sake, it is generally perceived that draft breeds are "quiet".
Likewise, it wouldn't really be ideal to have a "sleepy" Thoroughbred if considering their main job... Their job is, generally, to race, so generally those animals which are quick reacting and sensitive to everything are going to be what is bred... Which carries forward those traits more often than not. This gives the "impression" that TB's as a breed are "hot".
So yes, breed does "matter", to a point.
I would not be looking at a full draft for a serious dressage prospect, for example. If one came across my path which had the qualities and talent of a dressage horse I wouldn't turn it down without checking it out first though.
I would not go hunting for a TB or Arab for a beginner horse... But again, if one with the temperment suited for a beginner came along, I wouldn't say no based on breed alone.
So really, breed gives us a way to narrow a search for the right horse down - but it cannot be used as an absolute.
When it comes to a lesson horse, I would rarely, if ever, look for a "competition horse" to fill the role of "beginner mount", at least not for jumpers or dressage. The reason being that usually horses who are true competitors also have a temperment which is just too much horse for a person just learning. They are often "unforgiving" of mistakes because they are usually trained to a finer point and of a temperment which is just a bit more sensitive than what tends to make an ideal lesson horse.
I choose a lesson horse, always, primarily on temperment. Let's face it, a new rider does NOT need a Grand Prix mount... They need something patient, quiet, well trained and forgiving to their early mistakes. They don't need to be fancy, nor especially athletically talented, just sound - they are worth their weight in gold based on their mind alone.
So my one bit of advice is... Decide whether you want a horse YOU can show and move up on, or if you want "a packer" to give lessons on who also has enough ability for you to show from time to time on, because it is very rare you will find one capable of being both. Trying to make a horse meant for competition fit into the role of "packer" will almost always end up in injury sooner or later. (with that said, I have seen plenty of RETIRED show horses go on to be good intermediate level lesson mounts, and a few also suit SOME beginners... But this is after their careers as show horses are over and they are starting to slow down a bit)