That button doesn't actually do what you think it does (although I think the name of it could be clearer so I can certainly understand why people think that!).
All it does is hide the service account from within the game itself. This might not sound particularly useful, but if you have a service account that doesn't currently have a subscription it will hide it so you can't accidently click it and get booted out of the game since it doesn't have a sub. It also stops you from getting any renewal/expiry emails as essentially the account is deactivated.
But after cancelling a service account the Mogstation page simply has a new button for "Reactivate service account" which does the exact opposite and you can start using it again straight away. If you still have gametime left then you can carry on playing again. It does not pause your gametime though. If you had 30 days left and deactivated for 10 days, you'd still only have 20 days left to use upon reactivating.
However, if you go into the main Square Enix account website and then select update account information and "Cancel Square Enix Account", that will delete your actual account and all service accounts under it.
SE FAQ links about these:
Cancelling service account:
https://support.eu.square-enix.com/f...&c=0&sc=0&so=4
Reactivating service account:
https://support.eu.square-enix.com/f...&c=0&sc=0&so=4
Cancelling SE account:
https://support.eu.square-enix.com/f...&c=0&sc=0&so=4