There's no limit with the buttons, although it would become unusable. You can change this by creating your own theme though and use a drop down or whatever you'd like.
Another idea is something we've discussed before which is to register certain email domains with a specific IdP. For example <user>@
corp.com is automatically redirected to
idp.corp.com. With the new authenticator SPI you could create this flow yourself and remove the password field from the initial screen.
You may end up wanting to implement an authenticator for this in either case so you can add a cookie to remember the last used IdP.
When you use identity brokering in Keycloak, Keycloak becomes the "Service Provider" in the external IdP, not the individual clients. So only the Keycloak server has to be registered with the external IdP.