Hi Will,
Thanks for posting the link. In the 12 player schedule that you have commented on the intention is principally to balance the columns which might be different courts or fields at some central location, so we are not really intending it for the scenario where each team has their own home ground.
Have you looked at
my Excel spreadsheet that you can dowload here. You need to be logged into the site to see the download link. If you use this in home/away mode then it will generate optimal schedules for single or double round-robins which alternate home and away games as much as possible. When I look at the 6 player schedule on your page, then I see it has 6 breaks, where a break is defined as any departure from a strictly alternating pattern of AHAHA or HAHAH, i.e. HHAAH counts as two breaks. The optimal schedule has 4 breaks, and the Excel file will make it for you. I can find the references in the mathematics literature if you are interested in the method.