This is a tough one to check the balance for, because nothing else quite like it exists in the game. I'll compare the power of this relative to a Celestial Warlock with Cure Wounds, because that is also a touch-ranged heal with a short-rest recharge rate. Keep in mind, however, that Cure Wounds takes 1 action, so the opportunity cost of using it is much higher, but Cure Wounds can be used on the same target many times.
How do they compare at level 3?
At level 3, a Warlock with 16 Charisma can cast Cure Wounds twice as a 2nd level spell per rest, for a total of 2*(2d8+3) HP healed. The average is 24 HP.
A level 3 Thief with the Healer feat (presumably by being a variant human) will presumably have companions with 3 hit dice, so will be healing 1d6+4+3 HP, or an average of 10.5 HP, per party member, per rest.
So if the Thief is using it on three party members per rest, it is already surpassing the Warlock in terms of total HP healed, and the Thief gets to do it using its bonus actions.
What about at higher levels?
At 17th level, the Warlock hits its spellcasting peak, being able to cast Cure Wounds as a 5th level spell 4 times per rest. With 20 Charisma, that's a total of 4*(5d8+5) HP, or an average of 110 HP.
The Thief heals its level 17 companions for 1d6+4+17 HP each, an average of 24.5 HP. Now, the Thief needs to heal 5 companions per rest to outpace the Warlock. At level 20, the Thief is healing an average of 27.5 HP, and with 4 companions the Thief heals 110 HP- exactly the same as the full-powered Warlock. 4 PCs is the recommended party size in 5th edition, for reference.
How does it compare to other healing spells?
As another comparison, take Mass Healing Word, a spell that heals up to 6 creatures for 1d4+spellcasting modifier each, as a bonus action. It's similar in many ways to the Thief's use of the Healer feat, in that it heals every member of the party just once, and the bonus action makes it useful in combat. It's a 3rd level spell. Being generous and giving the 5th-level caster a 20 in their spellcasting stat, that's an average of 7.5 HP per creature.
On the other hand, the Thief can heal each of those creatures 1d6+4+5 HP (average 12.5) every short rest. Sure, Mass Healing Word delivers all the healing at once, but the caster forgoes a spell slot and can't cast any spell other than a cantrip on the same turn.
There are some caveats for the Thief. There's the object interaction needed to pull out the healer's kit, but that's fairly minor. A bigger cost is the literal gold cost of healing kits, which might limit usage of the healing enough to not make it ridiculously overpowered at low levels.
Finally, there's the limit of one heal per creature per rest. Having the option to concentrate the healing on one creature would make the option significantly more powerful, since normally it's just one PC who needs healing. Also, sometimes the Thief won't be able to get the full potential of their healing, when a party member manages to get all the way to a short rest without taking damage. This does put a major damper on the Thief's potential.
To conclude, allowing Fast Hands to work with makes the Thief a healer stronger than a Celestial Warlock at low levels, and comparable at high levels, and it's stronger than Mass Healing Word, a spell designed for the same purpose (whole-party, in-combat healing). The Thief gets to do all this with only minimal cost.
In optimal situations, this seems somewhat overpowered, particularly at low levels. But, the power of this option takes a big hit if healer's kits are less available, if damage is concentrated on particular PCs, or if some PCs aren't taking damage at all. If these mitigating circumstances are likely to happen in your game, it may be balanced, though still useful and powerful.