Yea, sounds like a good deal, just be careful that the “domain for free” isn’t tied to your web hosting service provider subscription! Will you loose the domain when you cancel the service? If so, then a “free domain” is much worse than one you paid for but that you directly own (via registrar). The whole point of “your own” domain is that you can keep it and the related email addresses regardless of email or web service provider.
Why don’t you just setup your own email server? Get a domain if you don’t already have one, it’s like $30/year? Then a small cloud server? Like 4 core/ 6GB mem, 100GB SSD for $5/month? Or use a home server? Then use Modaboa (FOSS) to setup a full email server for your domain, including trusted TLS encryption and all the current email security stuff (DKIM, SPF, DMARC). Yes, your email server will very likely be blocked from sending emails for some time, however, it works perfectly for receiving emails. I use it for wildcard emails so that I can just use a different email address for each service/website (i.e. unlimited email addresses) . Then I let google get my emails from my mailserver. If there is ever any problem with google then I can either directly get my mail via webmail or I just use another (free etc) mail provider or even a local thunderbird client to get mails.
Btw, backstory to setting up my email server is that I used to use the “free” email box provided with my domain registrar gandi.net . Then they suddenly wanted $5/month per mailbox and I said no and instead set up my own email server. That can happen to your service too, hence be careful that you fully own (including transfer rights etc) your domain.