Hi, ablesocial. I definitely understand why this would be important to you.
The checkout process is handled by the WooCommerce plugin, not The Events Calendar, so that’s why the fields aren’t helping.
Unfortunately, this is not really our area of expertise. The WooCommerce Tickets plugin allows integration between The Events Calendar and WooCommerce. Whatever options are available to you in the WooCommerce checkout should also be available for the event tickets you create.
That said, we do know that in the past people have had success with a third-party plugin that provides similar functionality. See this thread for details: https://tri.be/support/forums/topic/adding-extra-names-when-multiple-tickets-are-purchased/#post-46080
If you do decide to investigate it, please understand that we can’t offer support for it directly; you would need to work with that developer. Also, it was written for an older version of our plugin, so it may fail in unpredictable ways if it hasn’t been updated.
Hope that helps!