I've never heard anyone promoting the idea of keeping taxes complicated to avoid higher taxes.
There's nothing inherent in complicated taxes that stops the government from raising rates.
Conservatives have long wanted the flat tax (file taxes on the back of a postcard) and lower taxes, so simpler lower taxes.
I think the reason it became so complicated was because businesses lobbied for special rules and tax breaks that made things complicated in the first place. Businesses are now setup to take advantage of these loopholes (Amazon passing $0 in taxes, etc.).
TurboTax was built as a result of complicated taxes and they lobby heavily against a simplified tax code.
It's not some cabal of citizens wanting to keep taxes complicated, it's the industries that benefit from the current tax code that wants to keep it this way.