What bothers me is the jumping, not the falling. And not just jumping, but anything physical - swim speed, climb speed, wrestling down giants and dragons, etc. Without needing magic items or spells I think the fighter should be beating world records at 9th level, if not 5th. Not by a small bit either, but by the time the character gets to 20th, it should be as jaw-dropping as being able to take the above mentioned fall.
For example, would it kill the Fighter class if at 5th level they could double their Jump, at 10th triple it, at 15th quadruple it and at 20th have a x5 jump? Or the ability at 5th/9th they climb and swim at full speed (and at 5th they can swim in heavy armor without sinking - though I don't think that limitation is mentioned anywhere in the current ruleset...).
Just even these kind of things I think would be quality-of-life improvements for the Fighter/martials without even getting into crazy damage abilities.