TDM uses their own group checking I believe. It doesn't obey inheritance. This is one of the reasons why we advise against making groups 'higher' than superadmin since technically superadmin is always the highest group. Some 3rd party scripts may not behave properly.
You'll need to find the config for the TDM script and see if they have a list of groups that can pick up the cars. I've never used TDM so I can't help you there, sorry.
