I’ve seen pieces that have called the presidency a rather weak (and perhaps even almost irrelevant) office before TR and FDR reinvented it. Is this true, and if so, how was it weak? In what ways?
"Irrelevant" was never the case, but in the post-Civil War years a lot of it is the very vivid memory of Congress overriding 15 of Andrew Johnson's vetoes, which was an absolute humiliation of Johnson, as well as the memory of Johnson's humiliating impeachment and survival by one vote. Congress effectively castrating a president like Johnson got Congress used to being in the driver's seat. Add to that that a President's big task during the late 19th century was distributing patronage jobs to reward party faithful, not really "govern," and you get a bunch of weak, wastrel presidents. (No offense to Ulysses Grant or James Garfield, both fine men, but Grant was a weak President and we never got to see how Garfield would've turned out).
Add to that that a lot of the President's strongest powers on paper are in foreign policy and foreign policy was not a priority at this time and that's another route cut off.
After Grant and before McKinley, the most any President actually gets personally felt in the process of governing is Grover Cleveland issuing an absolutely staggering and comical number of vetoes, and even that is a more passive power and not an active one. (Would Grover Cleveland have to veto 170 bills if he had a better relationship with Congress? etc)