The engagement is one of the reason why they cut ties, while the demon slaying business is another, but Yuuto parent's greatest regret is the curse itself. Jinguuji side probably will lift it off once Yuuto parents agree with the engagement, but they don't and prefer to leave after realize the Amakawa head sided with Jinguuji.
Amakawa young family should return to the Household if:
- In dire situation for refuge, or
- When they forgive the past, or
- Agree with engagement or
- When the charm effect wear off
but they didn't make it in time
because their death in traffic accident
Amakawa family head already predict this, so they prepared Amakawa's family Heirloom, Yasutsuna, and hand it over to Himari, their demon servant, along with their command to come and protect their one and only last heir once the charm wear off, without reveal the engagement, to prevent the demon betray them and messed up the promise they made with Jinguuji Household.
The Amakawa's current head dead before the parents, thus leave Yuuto alone, unknown about the engagement promise and the protection put on him by other survivor Demon slayer house hold, till his power start grow stronger and overwhelm his charm that no longer capable to hide it's presence from the demon's radar.