Yes, you’ll get the alternating pile phenomenon more and more often. In reality, big “clumps” of cards from one pile or the other actually improve the randomization. In the extreme, this becomes the Faro shuffle https://en.m.wikipedia.org/wiki/Faro_shuffle

From a probabilistic standpoint, even if it isn’t a perfect Faro shuffle, almost-Faro shuffling clearly introduces far fewer bits of entropy which is less than ideal.