I would like to add my 2cp...
Having DMed for both a synthesist and a vanilla summoner, I found that although the regular summoner was indeed more powerful due to action economy, the synthesist was perceived (by me and other players) as more powerful: In my opinion, this difference between numbers on paper and perception comes from how the lack of weaknesses in the synthesist affects the DM when building encounters/challenges.
I'll give some examples (The synthesist was played levels 3-9)
-Through AC evoluitons + increasing DEX (increasing base + evolutions + spells/items) + mage armor + extra AC ('cause class), the syntyesist's AC was 6+ higher than the fighter, and that is not even counting when he had time to prepare for a fight and cast shield on himself. This disparity in AC for frontliners made it very difficult to choose appropriate challenges, since things that would hit the synthesist would hit the fighter with 8+ on the dice (which meant even iteratives hit), or you were left with enemies that could not hit the synthesist at all.
If this had been a normal summoner, I would feel less bad ignoring the pet, since at least I would still get to hit the caster.
- Through pounce + multiple attacks, the synthesist was getting full round attacks on round 1, while the fighter was running behind only getting one attack of (usually the combination of the 2 killed most non-boss enemies, so they had to move to the next enemy and once again only one of them got multiple attacks in. Like with the to hit, this made calibrating the enemies AC and HP very difficult.
I am aware that a normal summoner makes at least as much damage. But it was just the hopelessly outmatching of the fighter at fighting that made it feel so bad.
- Through skilled or evolution surge, the syntheisist could get +8 to any skill check. Combined with a high INT (thanks to placing STR and DEX at 8, which is low but not a super dump) and a more than decent set of class skills, he was at least as good of a skillmonkey as the rogue. And, since skilled allows to get +8 to diplomacy, guess who the party face was? I had to readjust all the diplomacy DCs or otherwise the whole adventure would have turned into "I ask the king for the key to the treasury. He agrees. I win"
In a normal summoner, you probably get as many skills, but it doesn't feel like you have an uber character that can do everything better than everyone, and the ridiculous +8 to skills doesn't come into play until later.
- Our cleric couldn't make it for a couple sessions. Guess who did not care at all, since they had crafted a wand of restore eidolon? Even after giving the rest of the party a wand of CLW, the synthesist only had to take care of himself, while the others had to share precious resources.
Before I am accused of being a bad DM (which I know I could do better)
- I was banishing the eidolon from time to time. But like it was pointed out before, there is a limited times you can do that without looking like you are just targeting one player (at which point I believe it was easier to just ask him to change classes)
- I was attacking them before they were prepared: The synthesists was actually one of the ones that handled this best, since he could still use summons and spells when the fighter didn't have an armor or the cleric his holy symbol.
I could go on (like how the synthesist can switch his immunity evolutiosn whenever a new level comes around, allowing the perfect counter to the elemental themed dungeon), but I think I have made my general point: For me, the problem with the synthesist is that it makes it very hard for the DM to make a balanced encounter where everyone has fun/feels risks, forcing either very easy encounters that will be completely overpowered by the synthesist, or very hard encounters that will make other characters feel useless.