Yes. The Primal app IS writing your notes to Nostr relays, even though it doesn't read from them. If you go to any Nostr client, your notes, comments, follows and followers will all be there waiting for you.
I encourage you to try out various Nostr clients. Each one does things a bit differently and has certain strengths and weaknesses.
For instance, even though I rag on Primal a fair bit, I still use their advanced search feature quite a bit, because it helps me find a particular note I am looking for really easily. Yet, when it comes to searching for a particular user, I always use Coracle, because I know that it will give me the legit user, based on web-of-trust, as the first result, whereas other clients may surface an impostor as the top suggestion. If I want to browse a specific relay's feed, I can definitely do that in Coracle, but Jumble is a better experience for that. Then for direct messages I stick with 0xChat, as Primal only does NIP-04 messages and while Amethyst does NIP-04 and NIP-17, it still seems to miss some NIP-17 messages.