Nah. I have hosted my domain for 17 years on google and then fastmail. The hosting is harder than private relay, although not too hard.
But I have only had maybe 3 services ever reject my domain, and those were because the domain contains a number.
Within the space of 2 weeks I had both Etsy and Mapbox block signups with Proton Mail aliases. The practice is rapidly becoming more common.
I've had some reject my e-mail address because it contains their company name. REI was one (ie it wouldn't allow [email protected] but would accept [email protected])