Thanks for using FusionAuth!
A couple of things.
You should be able to use Tailwind CSS to style the FusionAuth pages. You'll need to update your themed pages, but as I understand tailwind (based on a scan of this page) you can use it with any HTML pages, which is what the FusionAuth themed pages are. I'd suggest modifying the head helper to pull in the Tailwind generated CSS. What obstacles have you run into?
If you don't use the hosted login pages, you're missing out on a lot of functionality and security benefits.
But you can totally do that if it meets your needs. You can use the Login API or Password grant to directly post user credentials and get back a JWT.
One of the benefits of using FusionAuth is that your code (other than front end code) never has to handle user credentials, so that's why we recommend the Authorization Code grant.