I haven't read through this whole thread yet (I'll get around to it), but for now I can at least answer the OP's questions.
So, my question is three parts:
1) Do you feel D&D combat is slow (or "drags")?
2) If yes, how do you address this in 5E?
3) Has it always been that way? I'm not familiar with very much of 1E or 2E.
1) Yes, combat in 5th edition D&D drags on far longer than it needs to. It's repetitive, unexciting, and made worse by the tilt of the math in favor of nigh-inevitable PC victory. Anything that's tedious, time-consuming, and has no stakes is bound to be a drag on the game.
2) I address it by not running 5e at all and by playing 5e as little as possible, given the fact that I'm acquainted with numerous
Critical Role fans who all feel the need to "play like Matt Mercer."
3) In terms of combat speed, the difference between TSR D&D and the three WotC D&Ds is night-and-day. The first thing that needs to be pointed out is that even though TSR D&D has many versions (Original D&D, Classic D&D, 1st edition AD&D, 2nd edition AD&D), they all share the same basic "game engine" at heart and could be considered different games but aren't different
systems. Whereas each of the three WotC editions are all different systems. (The 3e/3.5/d20 System, the 4e system, and the 5e system are all incompatible with each other and with the TSR D&D system.) Each of these four systems has different characteristics which factor into the speed of a typical combat at any given level of play. TSR D&D is fairly rapid because there are few options to manage and small hit point totals. 3rd edition is similar enough to TSR D&D at low levels but becomes either a slow slog
or nuclear rocket-tag at very high levels. 4th edition is slow
by design because each encounter is "supposed" to follow the arc of players adjusting their tactics until they hit upon the strategy that leads to winning. And 5th edition is slow because of hit point scaling (bounded accuracy on the d20 modifiers… unbounded bloat on the hit point totals).