The specific sticking point seems to be that Jeffrey was on the panel of judges and won the award "Standards Champion 2010" whilst Happy Cog won the award "Design Agency of the Year". Nobody is claiming foul play. The debate is more centered around the value of these awards when Judges are also being nominated for and winning them. The other sticking point is the fact that it seems to be same group of people receiving nominations and awards every year.
These awards are mainly about web design. Design in this context, as pointed out by Oliver, is really all about communication and therefore those best at it will also be best at self-promotion. The awards reinforce this by magnifying that self promotion. The risk is that designers start designing with the aim of winning an award rather creating what the client needs. A bit like bad politicians enter politics for the power it brings rather than because they actually want to make a difference. Maybe there are other ways of measuring the success of a design project, perhaps through stats. But these awards are recognising quality not quantity.
An aspect of the debate is Jeffrey accepting the award "Standards Champion" for the third year in a row. I certainly don't begrudge Jeffrey accepting this award. He deserves it, although he personally has clearly stated that that award should have gone to Jeremy Keith. I suppose a problem is that this award is designed for Jeffrey. There is no bigger standards champion. Period. (Although Molly Holzschlag comes a close second in my opinion). Can't they just award him the honorary title "Lord of Web Standards" and from now on give the .Net award for Standards Champion to someone else?
One way of promoting lesser known designers for their work might be to have a subcategory in each of the personality focussed categories (Standards Champion, Design Agency, Web Personality and the podcast awards for a start) for up and coming designers. Perhaps this wouldn't be an award, perhaps it would be a nomination of five lesser known designers. I think this would be a good away of sayingt yes, we are picking the best, but we are also acknowledging others. And I suppose you should only be allowed to be nominated once in your life for such a "lesser known" designer.
Personally I think complaining about the same designers winning these awards is like complaining that the same tennis players are always winning Wimbledon. Maybe there is some kind of "natural selection" going on. The best keep on getting better because they can more easily surround themselves with high quality colleagues (be it usability designers or tennis coaches) to help them get better still.
Ultimately that is what is being awarded - continued self development and improvement in the field of web design.