Yes, Desciple of Life clearly triggers when using Healing Spirit.
Let's adress the rules first. Disciple of Life states 3 conditions:
- use a spell
- of 1st level or higher
- to restore hit points to a creature
Healing spirit is a spell of 1st level or higher that you can cause (use) to restore hitpoints. It almost word for word fulfills the conditions required for the disciple of life. This is how this feature works.
Addressing some concerns
It should not work because there is no action required to restore hitpoints.
Disciple of Life does not require you to use an action, it does not mention the action economy at all.
Also as @jgn pointed out, it is useful to note here that Desciple of Life does not trigger when you cast a spell but any time you use it to restore hit points. (This is somewhat unusual since many features trigger when casting a spell, but it is an important distinction)
It is not the character but the spirit who is causing the healing.
The spirit is not a creature, it is not referred to as a creature or anything that would have any mechanical in-game impact or agency besides healing and moving which you cause it to do. It is how the spell takes effect. You are not commanding a spirit do something, you are causing a physical manifestation of a spell to do it. If the spell summoned an elemental creature that could heal on its turn this argument would make sense, but that is not what healing spirit does.
A similar argument could be made for goodberry spell (you are not restoring hitpoints, eating the berry does), but it was clarified in the Sage Advice Compendium (SAC contains official rulings on how to interpret the rules) that Disciple of Life does, in fact, work with the goodberry spell.
Healing spirit is too strong.
In the end, it is the job of the DM to manage the game. If they feel the healing spirit is too powerful they can ban it or change it to be less powerful.
If this is the case, I would suggest changing the text of the spell to:
Healing Spirit "Once per round until the spell ends, whenever you or a
creature you can see moves into the spirits space for the first time
on a turn or starts its turn there, you can cause the spirit to
restore ld6 hit points to that creature (no action required)."
(highlighted text added by me)