I think the issue is somewhat more nuanced. The Constitution lays out a requirement that treaties be approved by a 2/3 vote of the Senate. Appointments, eg judges, only require the "advise and consent" of the Senate, implying that majority vote was contemplated. The filibuster is a Senate rule and hence subordination to the Constitution. The republican argument was that using the filibuster to block judges, ie requiring 60 votes instead of a simple majority, was inconsistent with the constitutional scheme.
There is no such objection to using the filibuster to stall ordinary legislation. That is what it was designed to do, to prevent the tyranny of the majority. There is a vast difference between blocking legislation and blocking the president's constitutional authority to appoint judges.
The Senate also adopted the so-called reconciliation exception to the filibuster for budget matters, recognizing that something crucial like the federal budget should not be held up by filibusters. The republicans used recociliation to pass the Bush tax cuts, whch is why they expired after ten years.
Obama's government health care plan is clearly not the type of bill that reconciliation was designed for. It involves the budget only in the same way any other big government spending plan does. That is the reason republicans are objecting so strongly to use of reconciliation. It amounts to the majority ignoring the Senate's own rules to push through highly controversial legislation. Of course, if we have learned anything about Obama, it is that he has zero respect for the law when it gets in the way of something he wants to do.