The inability for one house to influence the other in any meaningful, binding fashion has, as we see, the potential to create legislative logjams that result in the government being unable to move simply because both houses are not even voting on the same bills.
An amendment of this nature would not force a bill to pass, but allow a supermajority of one house (66%, the same required to override a presidential veto) to bring a bill to a vote within 10 days (the same period of time the president has to veto). This would give congressmen the opportunity to study and debate the issue, which is clearly pressing enough to unite the other house, while still providing a timetable to put bills to the test of a vote of all our representatives, rather than politically charged committees.