#Your house-rule does not address the core problem.
Why is healing unconscious teammates so powerful? It comes down to action economy. Forgoing a bonus action to grant an ally one or more turns is simply too good of a trade. Your house-rule does not change this fact and therefore it does not fix the problem, it just weakens the party in a roundabout manner.

I suspect that the use of spells like *healing word* and *mass healing word* will not decrease much, if at all. Instead, the party will just have to deal with the exhaustion aftermath.  

In fact, **your house-rule affects the social and exploration pillars more than the combat**, because the first level of exhaustion just weakens ability checks and leaves combat capabilities intact. Hard encounters will not become harder, but the rest of the day will be plagued with failed ability checks due to the exhaustion, which in turn will pressure the party into premature long rests and slow down the pace of the game. 

The house-rule also skews the balance of some character options.
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Characters who can cast *greater restoration* or *death ward* will have to spend slots to remove or prevent exhaustion.

Characters proficient in Con Saves (such as fighters), or that can improve saves (such as Paladins) are buffed. Meanwhile, characters that rely on ability checks (such as rogues, party faces, and skill monkeys) are nerfed.

Constitution becomes even more important than it already is.

Address the action economy directly.
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Rather than weakening the party with exhaustion, I would alter the action economy so that healing an unconcious creature is inefficient. There are many ways to achieve this, I'll leave you with two examples: "all bonus action healing spells/features use an action instead"; or "when an unconscious creature receives healing, it is incapacitated until the end of its next turn".

Either of these ideas will make combats more deadly, so you will need to adjust the difficulty accordingly. However, unlike your current house-rule, the exploration and social pillars won't be affected, and the only character options skewed balance-wise are healing spells/features.