I think it's going to far to call the study seriously flawed: The core claim presented "we find that superfluous apologies increase trust in the apologizer" is being properly controlled (superfluous apology, acknowledgment, or neutral greeting followed by a trust test).

Suggesting that the mechanism for that result is caused by ice-breaking effects, or by delaying the time between the introduction and the request, or by conforming to some unknown social norm are follow up experiments you'd want to perform... but there's always a follow up question, it doesn't make the initial discovery flawed.