The infusion fades and you must infuse the gem again
You already quoted the relevant part:
If it dies, it vanishes, leaving its heart in its space.
From the description about "Infusing an Item" in the Artificier description it states (emphasis mine):
Your infusion remains in an item indefinitely, but when you die, the
infusion vanishes after a number of days have passed equal to your
Intelligence modifier (minimum of 1 day). The infusion also vanishes
if you give up your knowledge of the infusion for another one.
So I would argue that if the text about the Homunculus Servant says that "it vanishes" you would have to infuse the gem again to create a new Homunculus Servant.
In addition the Steel Defender from the Battlesmith subclass reads similar to the Homunculus Servant considering the healing part (even if it is not an Infusion) but it expliticially states that it can be revived:
If the mending spell is cast on it, it regains 2d6 hit points. If it
has died within the last hour, you can use your smith’s tools as an
action to revive it, provided you are within 5 feet of it and you
expend a spell slot of 1st level or higher. The steel defender returns
to life after 1 minute with all its hit points restored.
So I think this is an additional indication that the Homunculus Servant can't be revived and needs to be infused again.
And since (in my opinion) the infusion vanishes healing spells won't work to revive it. The Mending cantrip states:
This spell can physically repair a magic item or construct, but the
spell can't restore magic to such an object.
Many other healing spells heal it while your Homunculus Servant is alive (except Lay on Hands for example since it doesn't work on constructs) but also won't bring it back to life, since the infusion faded.