Well here's the part that I mean.
A typical character can move 60 feet a round if they put their all into it (dash, move). A rogue can move 90 feet a round (dash, dash, move). A 2nd level monk can move 120 feet a round (dash, dash, move). A 5th level barbarian can move 80 feet a round (dash, move).
You gave an example of 96 ft/second, or 567 feet per round ... so I don't think that was right. 880 yard (2640 feet) in 2:45 (165 seconds) is 16 feet/second, or 96 feet per round for 27.5 rounds; that's a lot of dashing, far more than 3 rounds.
World record 100 meter dashes are under 10 seconds. 100 meters is 328 feet. 328/10 = X/6; that's about 200 feet per round (Usain Bolt's record comes out to 205 feet per round for 1.5 rounds). His 100 meter is 9.58 seconds, and 200 meter is 19.19 seconds (3 rounds is 18); that's hardly any slow down (his 400 meter is 45.28, or 26.09 seconds to run the extra 200 meters, which is 150 feet per round for another 4+ rounds ...)
Yes, this is mostly straight running or gradual curves, so a "run" action could have a "straight line" or some other limitation (1 turn every 6 seconds would make a pretty wide turn). Yes, these are numbers from a completely unencumbered person who has trained their whole life; monks don't wear armor, this should be doable by them before they start getting obscene speed boosts. Double dashing is not unreasonable given what people can actually do; D&D characters need to be able to be faster if they need to.