Red Mage does not do more damage than Summoner.
https://www.fflogs.com/zone/statistics/29#dataset=99
I've chosen the 99th percentile to eliminate the discrepancies between player skill levels. Whether it is raid DPS or adjusted DPS Summoner comes out on top of Red Mage.
Edit: Also, in more general terms, the "X is relatively more complicated to play and therefore should do more DPS" is really an archaic design philosophy. To start: in no modern MMO is any spec "complex." There are just not enough buttons to justify such a stance. The game is more diverse and enjoyable if every spec is competitive, and not because spec X has a few more oGCDs to weave.