This "trickle down" opposition goes back a lot further than Reagan. It actually began in the 1920s according to Thomas Sowell.
I'll stop the quote quoting in the interest of space and readability, but there is actually sound evidence that cutting taxes does in fact tend to raise tax revenue. When you slash the tax rate, people shift their wealth and income away from tax shelters like municipal bonds and are more likely to invest in taxable investments in the private sectors. Those investments tend to create opportunity and jobs. In 1921 the tax rate on those who made over $100,000 per year was 73%. By 1929, the government had cut the rate to 24%. During that time, tax revenue went from $700 million in tax revenue in 1921 to over $1 billion in 1929. Those with incomes over $100,000 paid 30% and 65% of those amounts, respectively. It seems counterintuitive to expect an increase in revenue from a reduction in the tax rate, but that's exactly what happened.
Woodrow Wilson seemingly advocated for the tax cuts during a speech to Congress in 1919 following that same logic.
Most notably, John F. Kennedy, the darling of the Democratic Party stated it outright: “It is a paradoxical truth that tax rates are too high today, and tax revenues are too low and the soundest way to raise the revenues in the long run is to cut the tax rates….[A]n economy constrained by high tax rates will never produce enough revenue to balance the budget, just as it will never create enough jobs or enough profits.”
Reagan was merely following the examples of those before him. The "trickle down" theory has nothing to do with wealth redistribution. As Thomas Sowell put it, it's just an argument against the sound economic theory, with empirical data to back it up, of cutting tax rates in order to change the investment behavior of the wealthy and generate more tax revenue.