Partially disassembling Sticky Shimano STI shifter for service
So, after studying the tech doc I took the plunge. It is actually not difficult and the only tools you need are some hex keys. I managed to successfully remove the lever assembly from the shifter for cleaning. Just don't try to disassemble any further than the doc instructs.

The bad news was that even after a thorough soaking in degreaser (which turned black from the grime) and relube, shifting performance has hardly improved, suggesting some mechanical issue. I think some new shifters are in order.

Reassembly for almost as easy, the only slightly tricky bit was getting the return spring in the right place without the official tool. A bit of poking with a pointy implement solved it though.

Conclusion: definitely a viable and effective way to clean shifters - if grease and grime buildup is the cause of stickiness.