here are some implementation practices -
the form processing code and form should be on the same page. by putting them on separate pages, you are creating a lot of extra code. by only validating one input at a time and not having the form fields 'sticky', you are providing a poor User eXperience (UX).
by storing the 'login_attempts' and 'lockout_time' in session variables, a nefarious user/bot can get unlimited new login attempts by simply not propagating the session id cookie between requests. you must store this data persistently on the server in a database table.
the only user related value you should store in a session variable upon successful login is the user id (autoincrement primary index.) you should query on each page request to get any other user data, so that any changes made to the user data will take effect on the very next page request, without requiring the user to log out and back in again.
the way a 'remember me' operation should be implemented is that if the remember me checkbox is checked, at the point of successfully verifying the user's credentials, generate a unique token, store that in a cookie and in a database 'remember me' table that also includes the user id, and the current datatime, for a determining token expiration. on any page request, if the remember me token cookie is set, query to find a matching row in the remember me table. if there is a row and the token is not timed out, use the user id from that row to set the session variable that identifies who the logged in user is. the rest of the code then uses this value in the session variable, just like it was set in the login form processing code.
the registration process, unless being performed by an administrator, which your code is not doing, should not include the role. the role should not be something that the user can decide when they register.
modern php (8+) uses exceptions for database statement errors by default - connection, query, prepare, and execute. any discrete logic you currently have testing the result of these statements should be removed since it will never get executed upon an error.
both the username and email must be unique or you should only use the email and forget about a separate username. the correct way of determining if a unique value already exists in a database table is to define the column(s) as a unique index, just attempt to insert the data, and detect in the exception catch logic for the insert query if a duplicate index error (number) occurred.
any form processing code should keep for the form data as a set, in an array variable, then operate on elements in this array variable throughout the rest of the code. i.e. don't write out a line of code copying every $_POST variable to a discrete variable.
you need to trim ALL the user supplied inputs, mainly so that you can detect if all white-space characters were entered, before validating the data.
you need to use an array to hold user/validation errors, and validate all the inputs at once, storing the errors in the array using the field name as the array index. after the end of the validation logic, if there are no errors (the array will be empty), use the submitted form data.
in the login validation logic, all you really care about is that the required inputs are are not empty strings, after being trimmed. by providing additional feedback to a nefarious user/bot, you are helping narrow down the values they need to try.