One reason, I think, is that it is generally the level of play that determines the level of fun, not the mod - even mods that are often called 'noob' (e.g. twinswords) could be fun when played at a more advanced level, and even Lenshu becomes somewhat boring at coloured belts (the public Lenshu tournament even had to remove grabbing). Mods like Lenshu and Aikido are therefore more likely to be fun due to their popularity with more experienced players, which, in turn, increases their popularity with experienced players even more. The cycle spirals upward and a large player base emerges.
Personally, though, I like Aikido because:
1. Timer
The timer intervals are very well designed. Each interval is short enough for creativity, variation, elimination of guesswork, and reaction, but long enough to simulate the limit of real-life human reaction time, keep the match exciting / dangerous, and allow for setting up powerful strikes.
The timer interval increase throughout the match is also well-designed.
2. Variation
Aikido and Lenshu, I think, provide for the largest range of viable play styles.
Big Dojo helps by further expanding this range to include more aerial styles. I have played / seen some pretty Lenshu-esque matches in Big Dojo.
Despite common complains about lifting / grabbing /etc., they, too, are valid styles of play.
The opener for Aikido is mostly irrelevant - spacing the first turn is totally viable. This is a result of timer settings and contributes to variation.