Breaking down the jersey before 1934:
In the pre-Eckstrom era, 1890-98, they initially wore the baseball uniforms but switched after a few seasons to horizontally striped scarlet and grey rugby sweaters. The "grey stripe" on the sleeve was born. They burned through a lot of coaches with Fred Ryder being the most prominent one, coaching from 1892-95, as well as 1898. They were a middling program that mustered only one Conference title before the OIAA disbanded. They only played Michigan once, in 1897, and it was a loss. They went only 3-6 against Kenyon, who was their primary rival at the time.
John Eckstrom coached the team from 1899-1901, which was the pinnacle of the pre-Big Ten era. They had an iconic look with Red sweaters that had a block O on the front and long, horizontally striped scarlet and grey sleeves. This is the inspiration for the modern jersey obviously, only with a block letter instead of block numbers. They won two state titles, including an undefeated 1899 campaign, kicked off the annual Michigan series with a tie in 1900, and went 3-0 vs Kenyon, which kicked off a winning streak in that series that is technically still alive to this day.
OAC era: 1902-1912. They rolled through this era with the Eckstrom jerseys, albeit with the caveat that only one player had the block O on the front; presumably the QB or maybe the head captain. The era was somewhat disappointing after the Eckstrom administration. They went undefeated against Kenyon, but went winless against Michigan with the lone tie in 1910. They did manage to muster a pair of OAC titles in 1906 and 1912.
Dr John Wilce era: 1913-1928. The Buckeyes joined the Big Ten in 1913, and Wilce guided the buckeyes from then until 1928. The Eckstrom jersey was replaced with the "Chic Harley" look that had leather helmets and a Scarlet sweater with vertical Grey stripes across the front. The grey striping was removed from the sleeves entirely. They stuck with that look the entire time that Wilce was the coach, with zero modifications whatsoever. A lot of innovations in that era obviously. Big Ten, Ohio Stadium, the first two all Americans, the Illibuck Trophy, Senior Tackle, and so forth. The Michigan series was suspended from 13-17, but Wilce collected wins over them in 19, 20, 21, and 28. Big Ten Titles in 16, 17 and 20.
The era between Wilce and Schmidt: 1929 to 1933. They reverted back to the old Eckstrom look for the jerseys, initially with numbered jerseys in 1929, but with the numbers removed for the duration of that era. Sam Williman was the coach, and he did not fare well. Script Ohio became a thing in 1932, so there's that. He did manage to muster a pair of Michigan wins in 29 and 31. The striping on the sleeve was dialed back from being a sleeve made entirely out of grey stripes to one that had just a small band of stripes above the elbow, more like the modern look.