Is “at turns” a familiar idiom as ‘at every turn’ ‘in turn(s)’ and ‘by turn(s)’ are?

As others have said, it's a general term conveying the idea of one by one.

I'd like to add, though, that at turns may have been used because there are two things that could be said to have come one after another: the candidates, and their gaffes. So, in no particular order: Bachman was perceived as ridiculous, Cain was perceived as self-destructive, Huntsman was perceived as ineffectual, etc.

The author probably didn't want to be so specific, partly because the correlation between candidates and gaffes isn't as clear-cut as I've indicated here, and partly because a single candidate may have compounded the said gaffes atop each other in a series of political missteps. So, an efficient way to convey this sentiment is to state the generality:

His declared rivals were at turns ineffectual, ridiculous, or self-destructive...

leaving the reader to work out the details of who stumbled when, and how each one stumbled. Meanwhile, the author can move straightaway to the main point of the sentence, which is that Romney had "a fairly easy, if occasionally fraught, path to the nomination."

I'd conclude that at turns is not idiomatic, but instead an efficient way to describe a sequence of people independently making a series of mistakes.

I think "at turns" is like one after the other.

I think the phrase "at turns" here has a literal meaning, rather than having an idiomatic meaning.

The following sentence:

His declared rivals were at turns ineffectual, ridiculous or self-destructive—granting Romney a fairly easy, if occasionally fraught, path to the nomination.

It simply states that:

Turn by turn (one after the other) his (Romney's) rivals were either ineffectual, ridiculous or self-destructive, which made his path to nomination easy.

British people sometimes talk entertainers having a turn. In a more sinister way, someone is said to have had a funny turn if they are taken ill or start to behave peculiarly.

The idea of 'turns' is quite straight forward. One person follows another.

In the world of entertainment, particularly Variety, the performers would perform one after another.

The idea of having a 'funny turn', is that you behave in a way that is not normal.

Fits, faints and funny turns:

http://www.racgp.org.au/afp/200304/20030413murtagh.pdf

Turns in the question, means 'as one follows the other'.

Here the phrase at turns means the same thing as by turns

one after another; alternately

It indicates that the rivals took turns being ineffectual, ridiculous, or self-destructive.

In English, we often see different prepositions being used in phrases but providing identical or nearly identical meanings.

The concept could also be conveyed by the phrase in turn.