I do not see what the problem is here in regards to restricting Turn Based to walking.
It is called Turn Based since everything is taken in turns, and in each individual turn there are "steps" being taken.
Walking would be counted as a "step" in a turn. To make this easier imagine each step as an action point.
Walking makes sense in TB since one step of movement is equal to one action point: 1S = 1AP
Running, however, just makes this formula complicated. How many steps do we take when running? The same as walking yes, but what about the required effort? It would not just be 1 AP since that is how much it takes when walking, therefore you will need more AP to run. So for argument sake let's just say it is 2AP to run.
To try and justify this concept of running in TB, let's say that running, costing 2AP, will let you move two hexes instead of just the one. However this is just useless since walking would get you the same distance at the same AP cost:
Walking to 2 hexes:
2 steps, 2 action points
Running to 2 hexes:
1 Step, 2 action points
Running is simply unnecessary in TB, and implementing it would be nothing more than a hassle. If you want to have the ability to run, then simply play in Real Time.
Real Time is where you always want to run and never want to walk.
Turn Based is where you walk because you are too afraid to run.