If we look at the total potential lifetime "clock cycles" of a system, and compare that number to systems that we know are considered intelligent and capable of thinking, it might shed light on the plausibility.
It takes information 100,000 years to cross the galaxy, and the universe is 14 billion years old (the galaxy is younger, but not enough to matter for this), so roughly a maximum number of 100,000 round trips could have possibly occurred in this "galactic brain".
Let's use a very conservative metric, such as how long it takes a human to blink in response to stimuli (about 100 ms), for our "human clock cycle". If a human lives for 80 years, that is about 2.5 billion seconds, or 25 billion clock cycles.
This is roughly 7 orders of magnitude (10s of millions) more than in the galactic brain. This gives the a very loose impression that a such system would probably have not have enough information flow or feedback loop cycles to "think" anything interesting.
Of course, this a very rough heuristic, but I think it is interesting and useful. The idea of unexpected or strange "thinking systems" is very cool and we should continue to explore it, but there are certainly some hard constraints defining the space of such systems.