As I pointed out in "If I undergo suffering now, will my next birth be better?" a while ago, liberation is not attained by "burning" past karma, but through "abandoning of unskillful mental qualities and the attainment of skillful mental qualities in the here-&-now"
As @Yam Marcovic showed in his "large boulder" quote (SN 42.6), Buddha of Pali Canon laughed at using rituals to improve someone else's rebirth. Indeed, how can your chanting help a dead person attain skillful mental qualities?! :)
However, Buddha never denied efficacy of rituals for changing one's own mood and motivation. In fact Buddha promoted usage of verbal aids for attainment of first jhana, and mantras can be seen as a kind of verbal aid.
In Tibetan Buddhism mantras and prostrations are used to cleanse one's own mind (not past karma!), one's own habitual tendencies. Specifically, mantras help stop inner dialog, and prostrations help against egoistic resistance.
From Mahayana perspective, denying efficacy of such self-directed rituals is foolish, because rituals obviously do change mood and motivation of the performer and participants.