tl;dr It takes the place of small potions for healing between encounters, and all but ensures the party will be at full hit points between encounters.
The healer’s kit costs 5gp and has 10 uses. That’s 5sp per use.
Better than Potions
A proficient character can be expected to restore more than a potion of healing for a fraction of the material cost, and has a reasonable chance to heal on par with a superior healing potion.
Better than Hit Dice
Offers comparable healing quantity to hit dice. Even an non-proficient character may get better healing than fighters or barbarians use of hit dice during a short rest. Given this can be done 6 times in the same span of time for 3gp, it essentially adds 6 nearly free hit dice to a short rest time span.
It is more flexible it when it can be used than hit dice. This rule can be used in short intervals when a full hour is not available.
Can have more inventory than hit dice. Hit dice per character are limited to their level. This house rule increases the healing capacity of a character to scale with the available weight and small amount of gold.
Removes hit points as an adventure resource to be managed
This house rule would eliminate hit points and hit dice as a limited or scarce resource in an adventuring day. When handling a half dozen encounters in a day, the choice of if and when to heal is a real concern.
Per the Dungeon Master Guide:
The Adventuring Day
Assuming typical adventuring conditions and average luck, most adventuring parties can handle about six to eight medium or hard encounters in a day. If the adventure has more easy encounters, the adventurers can get through more. If it has more deadly encounters, they can handle fewer.
Given even brief respite between encounters, the use of the proposed house rule allows a decently equipped party to inexpensively mitigate all damage between encounters.