As per the latest playtest packet, Bull Rush is a martial feat with the following text:
Effect: Choose a Large or smaller creature within 5 feet of you. Contest your Strength against that creature’s Strength. If you succeed, you push the creature up to 20 feet, provided you move with the creature along the same path. Doing so uses none of your regular movement, but you must be able to move.
As near as I’m able to tell, a Strength Contest has neither an action type or a limit to how often it can be repeated.
What limitations then, are there on Bull Rushing? Is there some reason I can’t continually bull rush against the same target? (Other than that would be silly, because I realize that is not RAI)
How is Bull Rushing supposed to work, or how can it be made playable and more sensible?
This has changed considerably since the playtest version. In the published D&D 5e rules, there is no “Bull Rush” option defined as a feat or action, in either the Basic Rules or the Players Handbook.
However, you can achieve much the same effect through grappling, as described on pg 195 of the player’s handbook. In fact, the mechanics entirely surpass the rules you listed.
You may only grapple a target no more than one size larger than you.
The grapple contest is Strength (Athletics) against either Strength (Athletics) or Dexterity (Acrobatics).
While you have another character grappled, you can still move, although your speed is halved and the character you have grappled moves with you.