You’re thinking of Teddy Roosevelt, not FDR. FDR was a Democrat. Teddy Roosevelt left the Republican Party, formed the Progressive Party in 1912, and ran against Taft, the GOP incumbent. Roosevelt split the vote, lost, and the Democratic Party candidate, Woodrow Wilson, won. The Progressive Party collapsed pretty quickly.
The U.S. has been a two-party system for the vast majority of its history – Federalists and Democratic-Republicans, Democrats and Whigs, Democrats and Republicans. Third-parties just don’t do well in this country. And whatever strong independent political parties did rise up were short-lived and collapsed quickly. The Republicans were the only successful example and that was because the Whig Party collapsed and left them as the only alternative to the Democrats.
Personally, I don’t tell people to vote or not to vote. Regardless of whatever I think of the candidates or the parties they represent, I ultimately think it’s the job of the politician to convince the public to vote for them, not shame them into doing so. But if you do choose to vote third-party (or, just not vote at all), you’re still ultimately helping one of the Big Two to win. It’s just how the system is.
