This cannot happen in the Parliamentary system, with a majority party forming the government. Disraeli's blunt " principles be damned. Support your party". still hold sway, as party discipline is firmly in place. The leader of the party has to sign the nomination papers of all candidates running for the party and the party whip, ensures that they toe the line when a vote comes up. Fusion of Legislative and Executive powers ( where the majority party form the government and the Prime Minister and his Cabinet ministers are chosen from that party and the members of parliament support whatever the Executive branch legislate.. This ensures stability in power and allows for clarity and consistency in governance ( it is clear what the government is doing and it is held accountable by the Opposition through debates and a Question Period and policies have to be consistent in order to avoid confusion)
The only time a majority government can be defeated, is by a vote of non-confidence and that can happen only if a majority vote against them. This would need a full scale rebellion from members of the governing party and that does not happen. Discipline is maintained ( political careers and ambitions are at stake) and stability is ensured.
Trump boasts of his ability to make a deal is farcical. He may have closed business deals. but this is government and the rules are different. Compromise is what marks a successful government ( even majority governments in a parliamentary system realizes that it has to make compromise at times, even though it does not have, because bringing people together is what a successful government does.) It is not about ego; it is about serving the needs of the people.
The " shut down" is the biggest and the latest of the many failures that mark the first year of Trump in office. He could not make a deal on the Muslim ban; he could not make a deal on the Wall; he could not make a deal on immigration ( except to threaten to deport the " DREAMERS"; he cannot find people of quality to serve in many vacant positions ( many diplomats are resigning ( the most recent is the ambassador for Panama) and so are many career public servants and some of those he has nominated have withdrawn their names because of poor qualifications and some are poorly qualified to do their jobs, with little or no experience or qualifications like the Education and Energy Sect. or even Tillerson ). North Korea continues to develop its nuclear program and all Trump can come up with is name-calling "rocket-man" etc. Foreign policy is in shambles ( recently NATO ally Turkey launch attack on America- backed and armed Kurdish rebels PKK ( HE GETS NO RESPECT) and NATO and EU has decided to ignore him and his erratic pronouncements.
China is rushing in to fill the vacuum, in Africa, Asia, building its Belt and Road ( the new Silk Route), while Trump fulminates. South Korea is making overtures with the North and as el talks and even got the North to sen athletes to the Winter games. Iran increases its influence in Syria, Yemen, Iraq and Lebanon and Russia rides high with airbase and port in Syria and remains in Crimea and Ukraine. Trump's one success is Gorsuch on the Supreme Court ( even his decision to move the American embassy to Jerusalem was roundly condemned by most of the countries, including all the major players.). All else is Failure, and his popularity and job performance keeps dropping, the lowest of any president in recent times. Trump has not made America great. He has diminished it. China is Rising. Russia has respect and North Korea rushes on.