Xeviat
Dungeon Mistress, she/her
My quick thoughts:
if you add in the run action go with this:
The run action cannot be combined with other actions, bonus actions, or movement.
When you take the run action, you move up to 4x your speed, but must move in a straight line. If you succeed at a DC 20 strength (athletics) check, you may move x5 your speed. If you are encumbered by weight or armor, drop the speed multiplier to x3 (x4 with successful athletics check).
This adds in the athletics you wanted, ensures no weird stacking, and handles armor encumbrance. I think this is all you really need.
I think this could be pretty close. I like the DC 20 athletics to go to x5 instead of requiring a "run" feat like 3E had.
I disagree that it's not needed. I outlined situations where it could come up. Chase scenes could last minutes, and the current move+dash mechanic doesn't allow for real human speeds. Urban campaigns may be dealing with smaller equipment loads too, not everything is a dungeon crawl. Long distance treks are the thing of heroic stories, so a team pushing themselves at 6 mph for 8 hours is doable; the "fast" overland speed only fits for heavily encumbered parties.
With the Mobile feat, 40 ft. movement is achievable. With a reasonable strength and expertise in athletics, a DC 20 athletics check is within reach. x4 movement on 40 ft. speed gets 160, x5 gets 200. We can probably say that Olympic Sprinters may be finding a way to squeeze another +5 or +10 movement rate. If a failed DC 20 athletics check also grants a level of fatigue, and if I made fatigue go away 1 point on a short rest, I could allow for these more. A runner can do 170 for a full minute by my findings, so that's the timer on x4 movement (160) before checks need to be made. So, here are my thoughts:
Move: speed x1
Dash: speed x1 (constitution save, possibly endurance check, every hour, DC 10+1 per check?)
Run: action, cannot take any other action or bonus action, speed x4 (constitution save every minute, DC ?)
Sprint: Athletics DC 20 for x5 speed.
I do also think I might modify the Charger feat and let charge be a regular bonus action with the dash action, but tie in some penalties with it. I know in the last adventure I ran, there were times where someone did have to double move to an enemy and not get an action, and as 3E vets they found this weird.