I like the thought there, but it only changes the issue a bit. Rather than healing up to full, the character just heals up to half. And if [MENTION=66111]Cybit[/MENTION] 's indication that this is more than just the Fighter is anything to go on, it won't solve the underlying issue.
Still, I like the way you're thinking better than I like the "if you want to ruin your game go right ahead" perspective on it. Some tweaks to the ability might just be enough of a solution. Making it regained on an extended rest rather than a short rest seems like a simple way to fix it to me, and could be applied to all similar abilities.
You or me or anyone else can do that without spending a dollar on 5e. Lets not imagine that this game is the only way that it is possible to do realize this goal, and so it is on every player to fix all the issues every time.
Which isn't to say that 5e needs to be the Perfect Game, either. It's going to have issues. It'll be more constructive to recognize those issues than to sweep them under the table, I feel.
For me, heals you can replenish with a short rest being more effective than a short rest creates an issue that simply saying "don't take more than one short rest in a row!" doesn't actually fix in any fundamental way. You need to nerf the healing abiltiies, or jack up the power of a short rest. Which isn't a table fix, it's a systemic fix. Not that table fixes can't patch it just fine.