Yes, but send and receive paths don't have to be symmetric. It's perfectly fine to send via the third-party provider and receive to your own server and forward

Yes, but send and receive paths don't have to be symmetric. It's perfectly fine to send via the third-party provider and receive to your own server and forward to the third-party provider. The forwarding MX would need to support things like ARC and DKIM, though, or the forwarded emails themselves run a high risk of getting dropped as spam by the third-party provider.