Email via web hosting service

Hi,

I already have a solution for private emails. Separately, I have another address bearing my name via mailbox.org, which I pay for; it’s not a whole, but I would like to limit my payments and the number of service I use.

Separately, for work, I use Strato.de to host my website and work email, which already bears my name.

That made me think I could use a custom domain I already own, and use that (contact at custom domain) as a replacement for my mailbox address. Less to pay, one fewer service, and no link with my private email.

But what’s the privacy downside of using a webhoster like strato compared to a dedicated service like mailbox?