I can't speak for UT2003 which I never played, but I whole heartedly agree that UT99 and UT2004 are vastly superior to UT3 (which may have killed the franchise altogether).
I played UT99 for hundreds of hours without ever playing a single Multi-Player session. UT99 and UT2004 are both excellent Single-Player games. UT99 includes a championship series for each of its game types. My favorites were Team Deathmatch and Assault (I was blown away when I first understood the historical reference of Overlord). UT2004 is also a super Single Player, my favorites being Invasion, Onslaught and Capture the Flag.
Multi-player multiplies the UT experience in every way. I can't begin to count the online hours I put in playing Invasion (UT2004). The number of Mods available was staggering. You could always find a site with players near your own playing level. Or just start one on your own.
UT3 did not establish a Single-Player experience. You can play Deathmatch, but the small dark maps were designed to look impressive-pretty; the UT Single-Player gameplay experience was abandoned in favor of the grand Mutli-player mystique. That was a mistake. I bought UT3, but I cannot sit and play any of it for more than a few minutes as a Single-Player. And the Multi-Player experience suffered for that.
Bottom line, if you play only UT3 then you will NOT have ever experienced Unreal Tournament.