The same with the presidency. It's even less democratic than the senate: the individual is not directly elected by voters. Once in office they have virtually unlimited power: executive orders, which require no approvals to enact. Only occasionaly do the courts step in to check them. No single person should have this much power.
The Constitution prohibits that even by Constitutional Amendment (seriously, the one thing an Amendment cannot do is mess with representation in the Senate), though it doesn't prohibit stripping the Senate of all of its functions and giving them to some new body.
You are not quite correct. Article V places two restrictions on amendments.
The first restriction has now expired. It stated that prior to 1808, no amendment could give Congress the power to forbid the importation of slaves, and no amendment could establish or grant power to establish a direct ("capitation") tax.
The second restriction is still in force, but what it actually says is that "no state, without its consent, shall be deprived of its equal suffrage in the Senate".
So equal representation in the Senate could be abolished by an amendment, if every state ratified it (since that would satisfy the requirement of each state giving consent to "be deprived of its equal suffrage"). It's also arguable that the 17th Amendment -- which changed election of Senators to be by the people of each state rather than by the legislature of each state -- already violated this clause by causing the Senate to cease to represent the states.
Would it be possible to amend the Constitution twice to accomplish this? First, amend the part that prohibits changing Senate representation, then amend the Senate representation?
Some individual rights need to be protected against the "tyranny of the majority." For example, a person with an unpopular view should still enjoy free speech. And of course, a majority should not be allowed to vote to enslave a minority.
That's a great argument for the 1st and 13th amendments. Perhaps those should actually be written in to the new document itself. We could even go further and adopt the Universal Declaration of Human Rights.
That doesn't change the fact that the American three branch bicameral system has not aged well and is profoundly undemocratic by design.