Your interpretation is correct in meaning, but speaking strictly, it's not a sentence but a noun phrase like "tradition - cutting off ~". Sorry for nitpicking.I think the sentence is something like that :
"This tradition is/is named : "cut off the tips for consolidate the origin" ."
It's "Sue wo tachite" since を shows that 絶 is transitive. I agree with the rest.Sue wo taete, moto wo kataku suru den