The U.S. seems to be the only western democracy where the upper chamber still seems to be co-equal to the lower chamber. In Britain, for example, the House of Lords is essentially close to useless, and so is the Canadian Senate.
Yet, the US Senate appears to be a co-equal to the House of Representatives.
Is there a reason why, unlike the other liberal democracies, the US Senate has not observed a decline in roles and responsibilities?
In a certain sense, America's founding fathers were terribly afraid of democracy, and went through great pains to mollify it with an extremely conservative system of government that deemphasized sheer population numbers. The people as a whole were feared for being fickle and unpredictable. The founding fathers did not want the federal government and country as a whole to change as a result of the current fads and emotional impulses. The House of Representatives, being directly elected by the people and in proportion to the number of people, would reflect that destabilizing fickleness.
The Senate, on the other hand, was initially designed to have its membership appointed by the State legislatures, not elected by the people. These were meant to be reasoned statesmen, who, combined with their much longer 6-year terms, could stand above the ephemeral fracas of the people and act in the greater and long-term interests of the states and the country as a whole. The fickle, fast-changing House was balanced by the slow and sartorial Senate. The same basic idea was used to rationalize the election of the President via Electors, rather than a direct, democratic vote by the people: the founders envisioned the electors as principled and intelligent men of strong moral fiber who could prevent the unfettered "will of the people" from electing a demagogue who gets by on vitriol and celebrity rather than an actual capacity to govern. Reality ultimately was much different, and electors these days are little more than puppets who are frequently penalized if they depart from the popular vote results, no matter the reason.
The States were also much more individualized, independent, and protective back then. None of them wanted to have the affairs of the entire country dictated by a small number of heavily populated states, whose affairs and priorities may be radically different.
As such the founders of the USA in fact endowed the Senate with substantially more power than the House had: they would confirm judges, cabinet members, etc. The only power the House has that the Senate does not is in the Origination Clause, requiring tax legislation to initiate in the House.
The Senate in the USA is an elected body. It can therefore claim a mandate from the electorate.
The House of Lords and the Canadian Senate are (mostly) appointed. The members of these houses don't have a direct mandate from the electorate. Moreover, there are deliberate constitutional acts to prevent the unelected chamber from overruling the elected on. It was the ability of the lower house to appoint members to the upper that allowed the lower house to force the constitutional changes that restricted the powers of the upper house. In the UK, in 1911, the then Prime Minister threatened to create hundreds of new Lords if the existing Lords blocked the Parliament Act which restricted the powers of the Upper Chamber.
In the case of the US senate, the senators have a direct mandate from the electorate, and as the lower house can't appoint senators, the House of Representatives can't pass constitutional changes that would limit the powers of the Senate by simply appointing hundreds of Senators, as constitutional changes need the approval of the Senate and the States.
The US is a federation of States; the purpose of the Senate is to maintain the balance of power between less populous and more populous states; as this purpose still exists so too its power does.
Another factor in addition to those already mentioned is that the U.S. has a strong Presidency, rather than a parliamentary system.
In a parliamentary system, you need to have a legislatively elected Prime Minister and cabinet to be chosen by and accountable to a legislature and it is much less natural to make a prime minister and cabinet (who exercise basically all real power in a parliamentary system) accountable to one legislative chamber rather than two. This tends to make the chamber(s) without a say in choosing a Prime Minister atrophy in power.
In a strong Presidential system, in contrast, the operation of the executive branch doesn't require strong legislative involvement or support, so, a bicameral legislative process is more feasible. A deadlock on any given important bill won't cause the day to day functioning of the entire government to cease. So it isn't as important functionally to make one chamber dominant and the other(s) subordinate.
The US Constitution sets out the specific roles and responsibilities of the US Senate, and those remain important.
Your two examples, the UK House of Lords and the Canadian Senate, are both composed of people appointed by the executive arm of government, and thus lack democratic legitimacy.
In the UK, this is the main cause of the decline of the Lords. The Lords remains somewhat useful, because the lack of reporting on it in the media discourages political posturing, and some of the people appointed to it have more expertise in useful fields than any professional politician. However, it is vastly less significant than the US Senate.
In Canada, there was a deliberate decision at the time of Confederation in 1867 to create a less powerful house to provide review and advice, and avoid the possibility of deadlock between two equally powerful houses. Making it appointed means that it can't acquire democratic legitimacy. Again, it is less significant than the US Senate, but that means it's working as designed.
1) Because it was set up that way in the US Constitution, so changing it would require amending the Constitution, which no one seems to want to do.
2) The system works, and arguably works as well or better than those effectively unicameral systems. If it ain't broke, why fix it? (Note that almost all US states also have bicameral legislatures.)
3) The US is not a "liberal democracy", it's a republic.
PS: I think a few comments are not quite what I meant by there being a difference between the liberal democracies of Europe, and the US "republican form of government". It seems obvious to me that the framers of the US Constitution were trying to restrict the powers of government while allowing popular/democratic input - the "checks and balances" so often discussed. One of those checks is the bicameral legislature, originally chosen by different methods, both houses of which needed to agree to pass legislation.
By contrast, the European liberal democracies seem to be constructed to allow unlimited power to the government, but stamp its acts with the imprimatur of democratic support.
