Hi Jeff,
I would not swap your 6 el. Yagi for any eggbeater (neither the original M2 one nor the homebrewed eggbeater II design).
I've build an Eggbeater II and it was very "nully" due to the many lobes of the antenna. I had it mounted about 20ft over the ground on the roof top and connected to a low noise GAsFET preamp. I removed it again after only one week of operation and was quite disappointed and went back to my homebrew ground plane.
Furthermore the antenna is only circular overhead and horizontally polarized towards the horizon. Even the simple 1/4 wave groundplane + preamp gave me a better reception of 70cm LEO birds at low elevation angles and much less nulls.
Generally I found vertical polarisation to work better most of the times with most kind of satellites (although by theory when receiving circular signals this shouldn't be the case - but you can centrainly confirm that with your yagi)
If you look for an omnidirectional antenna I would give the parasitic lindenbald antenna a try as described by Tony AA2TX in the AMSAT proceedings 2006 and 2007. I haven't tried it yet, but I expect it to work even better than any Eggbeater design (maybe someone ever compared such antenna types and could report here?)
Best 73,
Oscar DJ0MY