But.... carrying out the agenda of a president of the same political party would equate to and be the same as adhering to the wishes of their constituents.... the voters that put them into the majority.
Not necessarily. When someone is elected a senator they represent their state. That includes everyone in the state whether they are a dem or repub. Same with a representative. They represent voters in their district no matter what party they may be or whether or not they voted for them. When I call my senator or congressperson they do not ask me which political party I represent. My opinion should matter and I have a right to lobby my senators and reps and have them listen to me even if I am of a different party or didn't vote for them. They represent me and the rest of the people in my state. They owe their loyalty to us, not to the POTUS and his agenda. When they go to DC we expect them to answer to us first. That is where their main loyalty lies. That is why we call them our representatives. They do not represent the president.
This is the way it is supposed to work. There are times when a senator or congressperson votes against the agenda of the president because they worry about being re-elected or they have received pressure from those they represent.
When a senator is elected their constituency includes Independents, Republicans, Democrats, Libertarians and everything in between. In many states you do not have to even declare a party when you vote. In WI for example you could vote for a Republican for Senate and a Democrat for president. Some politicians run under a party not because they share an agenda but because they live in a district where running under the party that their beliefs most align with would not win them an election. Take for example Sheriff David Clark who is a Conservative Republican but runs as a Democrat.
This is the way our form of government was set up and intended to be.