Yes, both LROOE and GDIH are excellent songs, but they are very different kinds of songs.
LROOE is one of their epic 'serious' songs, and GDIH is just a good-time rocker. I like them both very much (but probably GDIH a bit more), but whereas LROOE is well-respected as a moody, epic, masterpiece of a message song, GDIH has been somewhat lost in the mix of great Eagles songs. So, for me, this is an easy choice, as I think GDIH is far more underrated than LROOE, and well, it is just time for LROOE to go.

Thus, I vote for Long Road Out of Eden.