So it is! With each halving, the share of new coins in the total volume is halved. Consequently, each subsequent halving has less and less effect on the market price of Bitcoin. Therefore, mining plays an increasingly smaller role in Bitcoin pricing.
So far the halving effects have always been short term and lasted for about a month or two causing a big hype in the market and then a big drop in that time. The bigger effects and the bigger rises that we see in what commonly referred to as "bitcoin cycles" is the result of growing adoption that happens for many reasons and will continue to affect the price.