Why is the notion of algorithm a primitive one in Brouwer's intuitionism?

As far as I know, Brouwer's intuitionism involves a primitive notion of "construction". This might be viewed as "a finite routine" --- finite because it should be possible to finish a construction. I think of these constructions or routines as somewhat nebulous things, not nearly as precise as what are usually called algorithms. So I wouldn't use the word "algorithm" for these things. But perhaps what Bridges and Richman have in mind is that, even though they choose to call these things algorithms, they need not be algorithms in the usual sense; the contrast with "sequence of symbols in [a fixed programming] language" seems to indicate something like that.

Since the work of Church and Turing (say around 1936), the notion of algorithm is definitely not considered primitive in intuitionism. But Brouwer started intuitionistic mathematics more than 2 decades before (1907, 1912), and in the absence of a commonly accepted notion of algorithm, he formulated his idea (somewhat) like this:

A sequence of natural numbers arises in the course of time, like a walk through the (infinite) tree $\mathbb{N}^*$ of finite sequences, choosing at each 'node' a next continuation.

The result is an infinite branch, or an 'arrow' as Brouwer called it. At any node (also right at the start) one may specify a definite finite law which completely governs all next choices (so that there really is no choice left at all). If one does so right at the start, the arrow is called a 'sharp arrow' or a 'lawlike sequence'.

Since the 1930's definition of recursion, it has become standard to identify 'sharp arrow' and 'lawlike sequence' with recursive sequences, I believe.