What you basically never see are two people standing in one spot smacking each other. Try that in a real fight and you're going to get hurt, a lot.
Nod. That's the 'static combat' issue that supposedly plagued 3e.
5e doesn't address it as voluminously as 4e did, but it does have a few things that help. THE big culprit in 3e was the Full Round Action (mostly the full attack, but there were full-round casting times, too). In 5e, the Move and Action are almost entirely separate. You can use your Action to Dash, but that's about the limit of the crossover. You can't 'sacrifice' your move for more oomf, so you're generally able to move, if you can find a reason to.
5e also softened AoOs, but that's a mixed benefit. It does make it easier to move around, but it also makes positioning less important in the first place.
Has anyone managed to come up with any minor house rules that encourage more in-combat mobility? Gives martial types good cause to move between attacks (even against the same target), or reasons to regularly fall back from an opponent?
I've seen DMs run 5e on a grid using essentially 4e rules for movement & positioning with good effect. Not always in the sense of a variant, either, just in the sense of not having internalized the dearth of such rules, yet.
Of course, 5e gave up movement/positioning rules and related combat options, in part to speed combat, and in part to default to 'TotM.' But, at the same time, it also reverted to giving speed/range/area in feet, even if it used the same 1":5' scale as C&T/3.x/4e, and spell areas in geometric shapes. So while you can run a melee combat in a fairly abstract way, TotM, once targeting an AE spell comes up, things can get weird. It's easy to be arbitrary when the underlying mechanics are vague & abstract, but when a mechanic calling for a very specific size & shape effect brought into it, there can be issues.
Of course, for us old-timers, those're issues we dealt with 30 years ago, at a 1":10' (or 30' outdoors) scale that didn't even match the 25mm minis the game was supposedly meant to be used with. (Up Hill! In the Snow! Both Ways!)