Regan did increase the debt, no doubt about it. Based upon your sig line/resume you are too young to remember. Regan cut taxes and the economy boomed. This is a fact. Now stay with me. To get what he wanted in terms of building up the military he had to compromise with Tip O'Neal (speaker of the house at the time) and we increased all manor of social spending.
You are right, Regan increased the debt due to spending, not due to government receipts (I don't call taxes revenue for the gubmint).
Tax cuts will grow the economy every time. BUT spending increased as well.
If you personally double your salary but quadruple your spending you're still in trouble. That in a nut shell is what DC has been doing for a long time.