I mostly agree with Harrison, Hayes and Arthur. I only bring them up because they got the job in our world, thus merit some discussion. But I do agree that while they deserve note, they shouldn't automatically get the jobs (assuming they all survived the war mind.) YOu're absolutely right about TR, but it's worth noting that he'll be interesting in the future.You make a good point on the cons of Lincoln going after a third term. I'm hesistant to support Grant getting the presidency considering what happened IOTL. Granted things are different ITTL but I'd like for something different then you know things happening like IOTL in case of Presidential Succession except better things happening. Especially with Hayes and Arthur, I'd rather that there be a different president like someone who never was IOTL. At the end of the day this is Red's timeline but I'd rather see different Presidents then you know just following what OTL Presidential successions were with just more successful results. Also Teddy Rossevelt is only a child by this point ITTL so his cahnces are definitely in the future.
The only note I'll push back on is Grant. He's got the name recognition, a (even more ITTL) sterling military record and now has a good working relationship with the Republican party. Basically all the reasons he got the nomination IOTL, but now tripled. And while his presidency wasn't perfect IOTL, he now has a chance to gain valuable experience in the lead up to a potential run. Maybe with him spending some time as an Ohio Representative? Either way, he's got a chance to go better than IOTL.
As I type this out, I'm starting to feel that Grant won't/shouldn't come immediately after Lincoln in 1868, but maybe the guy after that? Maybe one of Lincoln's cabinet gets the presidency, does a mixed job, and during those 4 years, Lincoln mentors Grant for a run in 1872?
Side note, imagine if Grant chose Frederick Douglass as his VP. Not sure it's possible or if he'd take the offer, but the image of Douglass in the White House is amazing.