The BTO made a number of points -

They were studies in woodland whose results are not necessarily translateable to garden birds (eg caterpillars are more abundant in woodland than gardens).

There were three studies in total, two on bluetits, one on gt spotted woodpeckers. The woodpeckers did better with supplementary feeding.

They were time limited, ie fat feeding during a short period only

They looked at productivity only, not overall success. Even if average broods were down, if more birds survived to breed, population overall would increase more

Their conclusion was "more study needed"

I don't know if this link is publicly available:
Are fatty foods bad for birds? | BTO - British Trust for Ornithology