It isn’t anti-intellectual to say that most of what the US federal government does would be better performed by private industry.
The problem is that industry is inherently narrowly focused in short-to-medium term profitability, and cannot be relied on to carry out work which benefits society as a whole, including many conventionally “intellectual” pursuits such as: educating the populace, or fundamental research with no clear path to monetization.
Yes, private schools do both of these things, but in both cases they are only doing so by means of public funding.