I would have thought that a party with a majority would not waste time seeking opposition votes. As long as your whips do their jobs and the members toe the line, you're going to get your stuff passed. Just get on with it!
In the matter of the filibuster, here in the UK, we have the 'guillotine' (allocation of time) or Programme motions which set out a detailed timetable of debate with votes forced at set times. This requires the Speaker to inform a filibustering speaker that they are out of time and that the House must proceed to a division (vote).
Then again, working with only two parties - the hard-right Democrats and extreme-right Republicans -you've got no clear space between the 'right' of the Democrats and the 'left' of the Republicans. This does not help in setting clear agendas or persuading voters that you have anything different to offer.
Until the two-party system is broken, and until a recognisable left appears in US politics, nothing will change.