You need to register the CPT with your custom capability, then assign that specific cap to the user.
When passing the arguments to register_post_type, set capability_type to your new capability, so the check turns into 'edit_cpts' instead of 'edit_post'.
By setting 'capability_type' => [ 'cpt', 'cpts' ], ‘cpt’ will map to ‘post’ for standard capability checks, so where you’d usually check for ‘edit_post’ you’ll now check for ‘edit_cpt’.
When you completely remove the primitive 'edit_posts' cap from users/roles as you’ve done, you’ll need to add in your new one with WP_Role::add_cap() and check for it with current_user_can( 'edit_cpts' ).