I mostly agree, but I think you can go all the way back to the years just after the Civil War and the symmetries don't look that outlandish:
Carter = Hoover = Garfield/Arthur
Reagan/Bush '41 = FDR = Cleveland/Harrison '23/Cleveland
Clinton = Truman = McKinley/1st-term-TR
Bush '43 = Eisenhower = 2nd-term-TR/Taft
Obama = JFK/LBJ = Wilson
Trump = Nixon/Ford = Harding/Coolidge = 2nd-term-Grant/Hayes
Again, these don't work perfectly, but if you squint hard enough you can kind of start to see a cyclical pattern.
I would argue you can go all the way back to 1824 with this particular cycle theory, though it works far better with the disjunctive and reconstructive presidencies than with the other presidencies, largely because of Democratic dominance from 1828-1856 and Republican dominance from 1860-1928.
These disjunctive presidents all left office extremely unpopular and gave way to new regimes:
Adams=Pierce/Buchanan=Cleveland (second term)=Hoover=Carter=Trump?
All these "reconstructive" presidents presided over a huge change in the political order:
Jackson=Lincoln=McKinley/T. Roosevelt=FDR=Reagan=2020 Dem?
These "first successors" largely upheld the party orthodoxy, but weren't as successful as their predecessors:
Van Buren=Grant/Hayes/Arthur=Taft=Truman=Bush
These "first pre-emptive" presidents had varying degrees of success in implementing their agenda, but failed to really change the underlying political dynamic:
Arguably Tyler=Cleveland (first term)=Wilson=Eisenhower=Clinton
These "second successors" were generally pretty successful at implementing domestic policies that adhered to the party orthodoxy:
Polk=B. Harrison=Harding/Coolidge=JFK/LBJ=W. Bush
These "second pre-emptive presidents" failed to change the underlying political order, but did point the way to the next cycle:
Arguably Taylor=No one=No one=Nixon/Ford=Obama?
I like the idea of using cycles to see parallels between different presidents. I think it would be a mistake to take the cycle theory too far and assume that all future presidencies will have to fit into the cycle in a predictable way.