Three spools is only one way of solving the compressor design point problem. Sure it's a great way of solving the problem and it requires an interesting bearing arrangment, but this in and of itself cant be the reason RR makes the most complex engines.

Three spools doesn't just solve a compressor design point problem, it solves a host of issues and typically results in a shorter, lighter, more rigid engine.

The cost of that is the staggering increase in complexity, it absolutely is the reason RR engines are more complex. Bearings barely scratch the surface. Lubrication, shaft failure safety systems and cooling/ventilation systems all increase in part count by a minimum of 50%, the number of modules increases which is more or less proportional to engine assembly/disassembly time, requiring multiple lifts of the stack that simply don't exist for two shaft architectures.

Consider that the Allison J71 and GE J73 were the first production engines to utilize VIGV's to reduce compressor stall during RPM transients, while P&W with the J57 chose to utilize a mid compressor bleed valve to solve the same issue. These were simple solutions. (These days everyone uses both).

I don't see what VIGVs and bleed valves have to do with 2 vs 3 shaft architecture given that every 3 shaft engine has both of these.

Ever wonder why the three spool system never caught on?

33% of aircraft that could have a 3 shaft engine do (or thereabouts), I'd say they've caught on. The reason GE and PW don't do them is they haven't historically needed to.

That brings me to my main point, when it comes to already complicated machines, the real brag is to find non complex solutions where you can.

That's lovely, but those days are long gone in gas turbines!

The RB211's competitors for example developed more efficient compressors to compete.

And that was great when the engines were new, but as the engines degraded those compressors became less efficient than the shorter, more rigid compressors on RB211s/Trents.

More recently, P&W designed the geared turbofan which effectively has three different rpm desing points while only having two spools. By putting is a gearbox between the fan and LP spools, the fan and LP spool operate on different RPM's but remain on the same %RPM.

They have, and it's still simpler than a full 3 shaft engine, because it doesn't have a hoofing big low speed turbine on the back end and 3 concentric shafts spinning in opposite directions. RR are going the same way.