I think the Senate would be fine if it was in charge of a Veto instead of having to also pass the legislation, also if it had a lot more senators to some multiple of 3 at a minimum.
IE doing nothing is just letting everything pass automatically and that cooling pan shit is something senate leaders have to pursue actively with (qualified) majority support.
My ideal procedure. House passes a law, Senate vetoes it with a majority meeting or beating the passing margin of the law in the house, but also representing a majority of all americans, house can override the veto by meeting or beating the population margin the senate's Veto represented.
You may note that there is no president involved in this process. That is because I believe the independent executive is an inherent threat to democracy and that it should be subject to complete erasure and power division to save the republic.