I just finished a project where we used F5 GTM's and F5 LTM's. I also am not an expert on LB this topic but from what I learnt...

The GTM's was nothing more that DNS RR, no real intelligence about load/sessions etc. The LTM's are making the smart decision's around least connected etc.

As a user logs in, GTM check down stream and passes the request to the LTM who then makes the decision of what UAG/Connection Server to use etc etc.

As sjesse stated, home sites are the way to sick people to locations or use weight the load balancer to the preferred site. Then you are running a active/passive what's not ideal for you I think.