From my reading on the filibuster, I get the impression that the filibuster in the US Senate today is equivalent to requiring a three-fifths majority. Why doesn't the Senate just change the rule to require a three-fifths majority to pass a bill?
Yes, they could, and this would probably be constitutional.
The rules of the House of Representatives include:
A bill or joint resolution, amendment, or conference report carrying a Federal income tax rate increase may not be considered as passed or agreed to unless so determined by a vote of not less than three-fifths of the Members voting, a quorum being present.
This has never been overturned in court and could presumably be extended to non-tax-related bills. So it seems that Congress can adopt rules that mandate a supermajority for legislation.
Edit: so why don't they? Probably because the modern filibuster mostly benefits people in the minority party, so the majority has little interest in making it more permanent. It only survives because some more moderate members of the party in the majority want to keep it.
You can't change the threshold of passing bills with house rules, that threshold is determined by the Constitution.
If you want to put a supermajority threshold on these things, you have to insert it in the form of another layer of procedure such as: "You need supermajority to bring the bill to a vote", which amounts to the same thing but does not contradict the Constitution.
It's worth noting that these mechanisms - arguably - are not real supermajority, since house rules can be changed with a simple majority any time. These hosue rules are only as strong as the norms which bind them in place. In this instance, the filibuster is strong because there is a strong institutional norm which holds it in place.