The claim you're making is itself an example of dishonesty and anti-intellectualism. Let's take a look at what Gove actually said - the full quote, not the abridged version people like to throw around:

"I think the people in this country have had enough of experts from organisations with acronyms saying that they know what is best and getting it consistently wrong"

Gove was clearly talking about a very specific subset of all possible 'experts' here - the ones from various (mostly governmental) acronymed organisations who have a long track record of making incorrect forecasts and yet still claim to be able to predict the future using their 'expertise'. The campaign was full of absurdly precise predictions like "leaving the EU will cost each household exactly £4300 a year".

Gove's position is not unreasonable or anti-intellectual. Post-vote, Paul Krugman started laying into economists and the economics profession as a whole, saying essentially that economists didn't deserve to be trusted because they so often made arguments that were just intellectual-sounding nonsense.

http://krugman.blogs.nytimes.com/2016/06/30/the-macroeconomi...

http://krugman.blogs.nytimes.com/2016/07/02/more-on-the-shor...