I think you're right that just quietly solving all the problems is not a great strategy for an employee, however I think this bit:

> Communicating this effectively is far more important than actually doing it, so I'd recommend spending 90% of your working time on this communication, and no more than 10% of your working time on actual problem solving.

...seems out of whack. We've probably all worked with someone like this, who did very little actual work and spent all their time on self-promotion and political maneuvering. They make awful teammates and bring everybody else down.

It's definitely worth advertising when you do solve problems, of course. If you hit a tricky one just send an email (either to team or just to boss as appropriate) with a quick description of the problem and how you solved it. No point doing good work if nobody's going to notice.