True I don't *need* a hard coded rule, but I do like to be consistent for my players and value the community hive mind for views as to how balanced this option might be, or alternatives that have been tested at other tables.
Plus it satisfies my touch of OCD.
Acrobatics is kind of overshadowed by Athletics, so it seems okay to want to let it cover this sort of thing. In contemplating a consistent rule that characters can avail themselves of whenever they want, allowing Acrobatics checks to increase speed, the question to my mind is how strong it is. Capping it to Constitution bonus is a good control on that, but let's think...
Rogue
Dex 18, Con 14, expertise Acrobatics, 5th level = +10. Short Rest abilities are frequently available every combat, and combats for a 4-character party take around 5 turns. In this case, they probably go for +50% and gain 30' movement per combat without risk. On top of Cunning Action, one impact this will have on your campaign if you use figures, is a need for much bigger battle maps! Seeing as your Rogue can Dash + Dash + Move*1.5 = 105' move on a turn. When they get to 11th level, with Reliable Talent, they perpetually double their base speed (skill check is +12 and minimum roll is 10). They can't fall prone, either adding 50% or 100% to speed, as they can't fail by 10 or more.
Monk
Dex 18, Con 14, Acrobatics, 5th level = +7. You'll need to rule if "base speed" is speed with or without Unarmored Movement (same for Barbarian). Given what Rogue is getting to do, it seems to me more just to say Unarmored Movement is adjusting "base" speed for your purposes. With Step of Wind a Wood Elf Monk can Dash + Dash + Move*1.5 for about 155'. They can't fall prone aiming for +50%, as they can't fail by 10 or more.
Druid
I won't analyse this, suffice to mention that Druids can turn into things with high base speeds.
Wizard
With Fly, what is my base speed?
I really have three comments to make
1) Your system isn't crazy: you've built in some reasonable limits that will stop applying to some characters at higher levels
2) There are pursuit rules in the DMG, that use Constitution to regulate Dash thus impacting who can catch whom
3) Many characters in 5e already have movement abilities that allow them to separate themselves out if a chase
One solution to your problem - how to separate characters out in a chase - might be first look at the DMG and see if that does the job, and second consider using Acrobatics to avoid being slowed, rather than to boost speed, as clearly a consistently applying system is going to warp at higher levels.