Does allowing death saving throws after being stable break the game?

After you roll three saves on a Death Saving Throw, you are stable and do not make other death saves. However, one of my player pointed out that if they keep rolling death saves, they might get a 20 and become conscious and be able to participate in combat again.

I’m planning to implement this rule:

After being stable, optionally you can roll a death saving throw on each of your turns. On a 20, you gain 1 HP and become conscious. Failing death saves does not destabilize you, including rolling a 1.

I don’t expect this to change things much. My goal is to give characters a chance to participate in combat again without being healed by others (we don’t have any healers other than a bard, who focuses on buffs/debuffs instead).

Are there any problems I should watch out for when implementing this rule?