I sit on a fair number of horses for an ammy rider, and it is always fairly immediately obvious which horses are regularly schooled in the walk and which ones use walk as simply the puttering/taking a break gait. Generally horses who aren’t regularly schooled in the walk have a super tough time marching up into contact and either get very fiddly in the mouth or back waaaaay off the forward when you pick them up - or they want to pick up trot immediately because it’s easier for them to push into the hand at the trot.
I think for h/j riders this may have less of an impact overall when it comes to successfully showing and even moving up than for folks looking to show in dressage where an unschooled walk is going to keep you at first or below pretty much indefinitely*. That being said, schooling the walk is massively beneficial for every aspect of training, with the added bonus that there is so much less wear and tear with a slower gait. Also, I would think that as courses become more technical and demanding, the finesse you can develop through schooling walk would be critical.
When it comes to walking simply for fitness, marching out on hacks really can’t be beat - both for fun and benefit to the horse. Long, forward walking hacks are great for horse and rider brains and bodies.
*I am not an h/j rider, many apologies if I am dead wrong.