The three branches of power in the United States government nominally serve as check and balance against each other, presumably keeping one from getting stronger than the other, and within each branch there's a variety of power: the Senate's filibuster, the varied levels of power of the Speaker of the House, the rise of the so-called imperial Presidency, the shifts in the Vice Presidency's authority, the changing regulatory power of the Executive Branch, the Supreme Court's activism or inactivity, and the nominal balance of power between the States and the Federal Government.
The challenge I set before you is as follows. Given the following statements as true:
1. The POD can be no earlier than January 1, 1900
2. The United States Government must still exist and cannot have fallen
3. No non-OTL constitutional Amendments can be passed
Change the way the United States government works, dramatically. Adjust where power and authority lie. Make the President a mere figurehead and the true authority lies among the Governors of powerful states, or the Speaker of the House rules the chamber with an iron fist. The Supreme Court can be a rubber stamp, or it's well known that it's not worth passing a law because they'll strike it down. The Vice President exercises considerable authority -- or has become so irrelevant there's no longer even an office at the Capitol for him.
Just sketch out a plausible POD and a chain of events that leads to a different structure of authority.
These don't seem like they'd be too hard to make. They all require different political conventions and established patterns, which can be changed by different personalities and dynamics. Lets take the elements one by one:
1)
Make the President a mere figurehead and the true authority lies among the Governors of powerful states
This one doesn't need any changes to the federal government, but to the states. What we need is a way to secure lasting and significant power for the state governors, and then find a way for that to translate into Federal power. The first will be a matter of creating very strong executives in the most populous and wealthy states. That doesn't seem difficult, because a number of states have gone through periods of having strong executive branches-Virginia comes to mind as one such example, and in this scenario Virginia will be one of the powerful states. So you could really pick whichever states you think will be most influential under this system and find a way to strenghten their executives. I'd pick Ohio, New York, Virginia, Kentucky, and California (if it's in the US at the time of our scenario) as the biggest, with Massachusetts, South Carolina, Mississippi, and Illinois as the second tier. The second part is linking the power of the governor in the state, to their power federally. Here, I think the best way will be to have widespread adoption of the South Carolina system of having the state legislature choose the state's electors. Combine that with a powerful state executive, and the stae's governor could have a great deal of power over the selection of electors and thereby over the way their state votes. You could add in other things too, like giving the governor a certain number of appointed representatives in the state legislature, and giving them a bigger role in choosing the state's senators (which at the time were selected by state legislatures) that would really emphasize the power of the governors. Lastly, I think all of these things would create a situation where the state governor was the dominant person in their state party. So essentially you'd be combining the powers of a strong executive (and remember, the state governors have the potential for greater executive authority than the Presidency) with those of a Tammany Hall-esque party machine, and then adding in strong influence over the selection of Electors and Senators. I think that would get you a US politically dominated by the governors of powerful states.
2)
the Speaker of the House rules the chamber with an iron fist
There have been periods of US history where this was true, to a greater or lesser degree. This one will depend on getting the right people in the right positions early enough that they can create the image and customary bounds of the office.
3)
The Supreme Court can be a rubber stamp
Just have a successful "court packing" bill at some point. Perhaps under Jackson? That, and have Congress strip the Court of all but its constitutionally defined jurisdiction. Most of what the Court reviews these days comes from Congressional expansions of the Court's jurisdiction.
4)
it's well known that it's not worth passing a law because they'll strike it down
This one is harder, and I'm not really sure how you'd do it. The Court has the potential for a great deal of power and independence of the other branches, but the problem is getting that to last beyond one judge's tenure. This is especially problematic with your injunction against ATL amendments, as that's really the only way to make the Court more powerful than it is already. The best way would be to have a relatively early, relatively activist Chief Justice who could expand the Court's reach and make that part of the customary role of the Court when those things were still being settled.
5)
The Vice President exercises considerable authority
Simply, have someone other than Adams as the first Vice-President. Adams was initially a fairly restrained VP, and when he tried to be more active drew substantial criticism. A different individual might be more successful in the same situation, and establish a more powerful Vice-Presidency.