Well, first I think you need to rethink your Z-axis design because that cannot possibly work how you have it shown now.?

Regards the Gantry height then it depends on what your cutting. All the machines I build have at least 100mm of Z travel and most have 150mm and will happily cut aluminum at full extension with a reasonable finish quality and accuracy using correct feeds n speeds. If you require a higher finish quality or deeper cuts or higher feeds then yes it's better, even necessary to have a lower machine or if not then a machine that is built much much stronger than your typical router needs.

To be honest I'm not a fan of the open gantry design you are using with the profiles orientated in the vertical position which is there weakest orientation. Those same two profiles could make a much stronger gantry if you made it an L-shape and put the screw at the rear.