In general we should expect a larger brain (ours consumes 1/4 of our total energy) to pay for itself in more actual intelligence.
I don't think we should generalize from the single data point of recent human evolution. Are we sure larger brains tend to give greater general intelligence in other lineages? Has this been checked? Is the intra-species brain size variation large enough to check this in species where we already run intelligence tests?
The fact that human brains recently became unusually large can be explained by other theories. For example, it seems more likely to me that first there was a (relatively recent) mutation that significantly changed brain architecture, and only after that point did the new brains profit from growing larger. (E.g., chimps would have come before this mutation, and indeed have not experienced runaway brain growth.) On this theory, brain size only benefits the lineage with a very specific neural architecture and isn't a general rule.
There have also been other theories, some of them invoking sexual selection.
if it doesn't (the architecture can't easily scale) then the smallest (still working) brains will win.
Most of the brain does tasks not directly related to general intelligence, e.g., lower-level input processing, managing the digestive system, etc.
Increases in brain size might also improve these functions and be an advantage in their own right. This would further muddy the issue if we charted intelligence vs. brain size, because we don't know enough yet to say which parts of the brain (particularly of a non-human brain) have which general-intelligence functions. Bird brains, for instance, aren't much like mammal brains, they don't even have a neocortex, so we can't just compare growth of brain areas directly. (Not to mention cephalopod brains...)
Finally, brain size directly correlates with head size, and most of the time I expect head size would be more evolutionarily significant. It controls things like eating (size of mouth and throat), defense (size of teeth and jaw), acuteness of the senses (eye size)...
birds lack the physical attributes to profit from any more general intelligence than they already have
Not to nitpick, but the definition of fully general intelligence is that pretty much anyone can benefit from it.
Is there any evidence that fully general intelligence exists in this world?
We have a sample of one modern human civilization, but there are some hints on how likely it was to happen.
Major types of hints are:
Data for:
Data against:
To me it looks like life, animals with nervous systems, Upper Paleolithic-style Homo, language, and behavioral modernity were all extremely unlikely events (notice how far ago they are - vaguely ~3.5bln, ~600mln, ~3mln, ~200k or ~600k, ~50k years ago) - except perhaps language and behavioral modernity might have been linked with each other, if language was relatively late (Homo sapiens only) and behavioral modernity more gradual (and its apparent suddenness is an artifact). Once we have behavioral modernity, modern civilization seems almost inevitable. Your interpretation might vary of course, but at least now you have a lot of data to argue for your position, in convenient format.