First: Teddy would get a better deal, and Teddy + Hughes would get their better deal through the Senate.
I'll leave to you all to work out the details (which you more or less have,
MarkA aside), but Senate Republicans aren't stupid and Teddy is way more popular than they are plus they have to live with Hughes for a few more years.
As for the election in 1924, it largely depends on Hughes' second term, IMO. Hughes will owe TR's Progressives big and so the conservatism of the Republicans may be blunted; I doubt it and Hughes' second term is probably fairly pro-business. Hence, it's not out of the question that a Democrat wins in 1924. Maybe Hiram Johnson though? It would require the Progressives having a bit more clout, but TR's legacy in *Peace might give them that. He also gives a nice geographic balance to the ticket.
Alternatively, you have Hiram Johnson manages to get on the ticket (in a way very similar to Coolidge in OTL 1920). His progressive has the chance to respond well to economic crisis, but so did Hoover OTL. In any case it'll be hard to get him relected after so long with Republicans in the White House.
Long term there's a decent chance for the Republicans to keep more Progressive influence than they did OTL if they have a more formal re-union and if they avoid the swing to the right in reaction to Wilson under Harding and Coolidge. I also find it particularly amusing to have a Democrat in power for a alt-crash in 1929, just for fun.
The Republicans, until the Great Depression broke them, pretty much had an electoral college lock. Sure they needed 2/3 of all non-South EV, but except for the 1912 split (and 1916 incumbent / looming war) and of course Grover himself they always won since the end of the Civil War.
I see no particular reason the Republicans would lose the Presidency anytime soon, although that takes us straight into their OTL problem: having the Presidency when the Great Depression hits.
It seems odd just to have Republican Presidents one after another, but before the Great Depression and after the Civil War that's basically what happened: 16 Democratic years (Wilson, Grover), 48 Republican years. (Or we could do like 8 progressive years (not always liberal) and 56 conservative years)
A Republican Party with Teddy Roosevelt as Secretary of State probably appeases the Progressives somewhat and it's quite possible that a proper mend between the conservatives and progressives could be formed. They do have more common interests than with the racist Southern Democrats (at the time, anyway) and they can probably agree on foreign policy and some domestic policy.
That said, if the progressives are participating fully in Republican politics that's going to push the party, not really to the left as such, but certainly in an anti-business direction.
Particularly, I think they demand (after the pro-big business Hughes) a progressive for 1924 at the top (pointing to increased progressive led Congressional majorities in the ATL & California's important to 1916).
Notably if Teddy lives a little longer he may be able to bring about not just OTL 1918 Republican congressional pick-ups but also 1920 ones—a strong ATL progressive force in the House will do a lot to the Republican Party.
Let's say, for the future of a Great Depression hit Republican Party, Hiram Johnson is the President elected in 1924 (with, certainly, a strong conservative VP).
(We could put the Democrats in charge when the Great Depression hits, but that would leave neither party able to react, both dominated by conservative forces. Although the New Deal may have lengthened the Great Depression (IMHO)—it also got people through it without a widespread break-down in society, or revolt against the government so it more than worth it. No New Deal of any kind… yikes.)
He's got eight years, a Republican majority or close to it (albeit also a conservative majority if you count Southern Democrats + conservative Republicans), and—unlike Hoover—he can communicate that he is in fact doing stuff about the Great Depression. So hey, the US might get another burst of progressive work. Could be interesting.
Although Hoover gets blamed a lot (try Coolidge), he really did react to the Great Depression—he just was never seen as doing so. Hiram Johnson would be seen as doing so, I think. Certainly he would have followed different, perhaps better, policies. If he had managed to get national health insurance passed (unlikely, but…) that would be a powerful cushion through the Great Depression & a political saviour for Republicans.
Now that's not enough to save whoever the 1932 Republican nominee is, but I could see the Republicans surviving the Great Depression. Not enough to win the Presidency anytime soon, but enough to stay competitive and retain decent Congressional forces (though, if it is FDR, the progressive Republicans will likely back him).