This gives you the benefit of disable-able email addresses, but not the benefit of privacy. Those companies (and once leaked, anyone) can use your custom email domain as a fingerprint for your online activity.
(Source, I'm the creator of Owl Mail [https://owlmail.io] and this is a common question.)
If you give your personal email address to hundreds of services online one of them is bound to sell or leak your email.
By using Owl Mail (or Firefox Relay, etc.) addresses everywhere, you reduce your attack surface to one security fastidious company.
And, even if Owl Mail (or Firefox Relay, etc.) were to experience a data breach, at least it would greatly increase the effort required to match emails to your identity.
Also, I think one thing you should look into as a natural evolution is promoting the use of auto-generated, secure passwords unique to each relay address.
I was able to set up alias emails in my gmail & have all emails from a particular domain forward to my domain as well.
Then went with a password manager & changed all my email addresses to my own domain with specific relays (amazon@ netflix@ etc etc)
Works really well for ~12/year!