I have said this before but I will repeat myself. When a rail route franchise is up for renewal a list of requirements will be listed. For Thameslink which covers Luton that list extends to over 600 pages. This requirement will cover the minimum service allowed for different parts of the day and night. If that list does not require early and late trains that benefits the airport or even overnight trains then it will never happen as these sort of services can have a low demand so run at a loss.
The airports only hope is to persuade whoever writes the requirement to add these extra trains to the list. Then when the rail companies do their calculations they can come up with a bid that covers all the trains required to be run.