I used to think republicans were for
christian values
strong military
law and order
stability and deregulation
low taxes and rampant homosexuality
With one exception, I think the above is representative of Republicans in general. As a Libertarian, I cringe when you refer to "rampant homosexuality." I would have replaced "rampant homosexuality" with "privacy." I want government out of our bedrooms and I consider sexual orientation to be a matter of private concern and should remain a private matter.
As for the rest, while I am an atheist in the sense that I see no evidence for the existence of a supernatural being that rules the universe and find the concept to be at odds with natural law, I have an appreciation for what is generally considered Christian values. In addition, I place value on the notions of a strong military to defend our freedom, law and order to defend our personal security and stability of our institutions. Deregulation does not exist as a first principal but rather a reaction to stifling overregulation by the liberal elite. The trick is to maintain sufficient regulations of our institutions for a level playing field while avoiding overreach of the bureaucracy.
With the exception noted, I consider the above list to be consistent with Republican values and the efforts of the Trump administration.
As for much of the rest of your post, it appears to me that you are attempting to project your parliamentary form of government onto our constitutional republic. In a parliamentary form of government, the parliament is supreme and the prime minister serves at the pleasure of parliament. And certainly your form of government has certain advantages. For example, you never have divided government in the sense that we can have divided government.
Nevertheless, I prefer our constitutional republic. In our constitutional republic, the President is chosen directly by the People at regular four year intervals and represents a separate branch of our government. Because of possible overreach by the executive during periods between elections, the Congress is given an oversight role. And in certain extenuating circumstances, the Congress actually has the power to remove the President from office. But, according to our constitution, this should occur only for specific crimes such as treason, bribery or other high crimes and misdemeanors. According to our Constitution, the reason for removing a President must be more than merely finding disfavor with the Congress.
In the present case, on a basic level, the Congress has charged the president with incurring disfavor with Congress. It charges no crimes. The Congress has issued two articles, Abuse of Power and Obstruction of Congress. Neither is a crime. As for Abuse of Power, it is not unusual for a President to be accused of abuse of power. The Constitution gives the President an enormous amount of power and Presidents are typically accused of abuse of power by someone. But abuse of power has never has been grounds for impeachment because it is not a crime. Neither is Obstruction of Congress. The executive has a right to appeal to the Judiciary. Pretending that either abuse of power or obstruction of Congress is a crime is nonsense.
What we are witnessing is an abuse of power, but not by the Executive. The abuse of power is a political play by the House of Representatives which is attempting to execute their role consistent with a parliamentary form of government--not a republic. What they are doing is unconstitutional and deserved judicial review.
Unfortunately, it does not appear that the Constitution has appropriate provisions for judicial review of Congressional misconduct. It appears, therefore, that in this respect, our founding fathers let us down. The President should have the right to ask for judicial review in the case of impeachment and, in my opinion, we need a constitutional amendment to give that right to future Presidents.