[3.P] The Summoner's Handbook: A Guide to the Pathfinder Summoner

[Barstro]

As for the pincers, you can have them as primaries if you don't use your bite/tail that round (though yes, I'd love to get rid of them if I'm not gonna use it):

I respectfully disagree with your interpretation of the phrase "If a creature has only one type of attack, but has multiple attacks per round, that attack is treated as a primary attack, regardless of its type."

You obviously take that to mean "If a creature uses only one type of attack".

My rules lawyering (sadly) takes it to mean "If a creature possesses only one type of attack"

Right idea, wrong Base-type.
Pincers:
Pick a Biped.
Replace Claws with Pincers.
Add more Arms+Pincers.
Do NOT add any other types of attacks.
Go to Pincer-town!

The replacement is actually a key ingredient in the "Zoidberg"-type Eidolon build.

You are missing the main part of my hypothetical pincer build.

Step 1) Attack with pincers (primary attack if I could get rid of the other unused attacks)

giving it two pincer attacks. These attacks are secondary attacks. The pincers deal 1d6 points of damage (1d8 if Large, 2d6 if Huge). Eidolons with the grab evolution linked to pincers gain a +2 bonus on CMB checks made to grapple.

Step 2) Successful attack creates a Grapple check

Pick bite, claw, pincers, slam, tail slap, or tentacle attacks. Whenever the eidolon makes a successful attack of the selected type, it can attempt a free combat maneuver check. If successful, the eidolon grapples the target.

Step 3) Use Constrict

Whenever the eidolon successfully grapples a foe using the grab evolution, it deals additional damage equal to the amount of damage dealt by the attack used by the grab evolution. This evolution is only available to eidolons of the serpentine base form.

Step 4) Release the grapple

you can, as the creature that initiated the grapple, release the grapple as a free action, removing the condition from both you and the target

Step 5) go back to step 1

This would allow the possibility of getting double damage of 1d6 for each pincer (only for serpent) as opposed to 1d4 and some rends for a claw eidolon.

But, as stated, I think that the build does not allow for the pincers to ever be primary. I also think that the eidolon could only ever have five pincer attacks instead of seven because the unwanted bite and slap take up space because "If the eidolon is at its maximum, it cannot take evolutions that grant additional natural attacks."

If there could be an evolution that gets rid of initial evolutions, then my hypothetical pincer build would be good. But, since there is not, pincers are stuck at 5-max secondary attacks.