...that BBC Sherlock went "James Bond," by having Sherlock work at Mycroft's (and therefore, the government's) behest all the time?

The problems with this:

1) There is a lot of good material in canon that they're missing out on, involving Sherlock solving - I don't want to say "ordinary" cases, because he likes cases that are out of the ordinary...but, more "private" or civilian cases. He does get a few national-security-issue-type-cases from Mycroft in canon, but they don't make up the bulk of his career. But they seem to make up most of the BBC series.

2) Central to the character of Holmes is his independence from institutions as well as people. In the series, it seems like he's working for Mycroft and the government.