I think the biggest problem with people here is that so many people on this forum seem to expect every player on the team to be an all-star, and if you aren't one then you're crap and should be traded.
Avery Bradley is a pretty efficient 15 PPG scorer, an excellent defender, and a solid rebounder. In the NBA, that's a starting calibre player.
The biggest criticisms people seem to have for AB are:
1) He's not a good ball handler
2) He's not a good passer
3) He has poor shot selection
4) He's inconsistent
5) He's undersized
Lets address some of these starting with points 1 and 2...
Passing and Dribbling
There's no denying that AB is not an above average ball handler or passer...but the fact of the matter is, he doesn't really need to be. He plays purely off the ball and rarely ever is the primary ball handler, so he doesn't have to have PG handles or passing ability. He's also averaged only 1.9 turnovers per 36 minutes over his career, so it's not like he is constantly dribbling off his foot or passing the ball out of bounds...these things aren't a liability, they just aren't an asset. If he could pass and dribble (in addition to his current rate of scoring, defending and rebounding) then he'd be pretty near All-Star consideration. As it is, he's a starting calibre player.
On to point 3...
Shot Selection
For his career Avery Bradley has shot 44% from the field, 36% from three and 78% from the free throw line. For a guard, they are really quite good numbers.
Statistically, the majority of AB's offense (34%) has come from behind the three point line, where he's shot 32.4% so far this season. Because a three counts as 1.5 two point attempts, that's the equivalent of him shooting 48.6% from two. That's pretty efficient.
The second largest amount of his offense (29%) comes on long two's - between 16 feet and the three point line. So far this season he is shooting a whopping 51.7% on those long two's which means for him, that's an even more efficient shot than the three pointer is right now.
The third highest number of his shots (26%) are coming from within 3 feet of the basket - i.e. layups and dunks. Obviously they are pretty efficient shots, and sure enough he's shooting 50% from that range - not fantastic relatively speaking, but efficient as a shot.
The two worst shooting areas for Bradley (over his entire career) have been the close jumper (3-10 feet) and the midrange jumper (10-16 feet). So far this season he has only attempted 6% of his shots from 3-10 feet, and only 5% of his shots from 10-16 feet. Look at his career stats and his attempts from these ranges have consistently hovered around there. He obviously acknowledges that this area is a weakness for him, and so he takes lets shots from there.
So to summarise, Avery Bradley has taken 86% of his field goal attempts this season either from three where his effective FG% is 48%, or on two point shots where his effective FG% is over 50%.
Where is the evidence of poor shot selection?
I can't see any evidence to indicate that AB is taking a large number of low percentage shots.
Inconsistency
This is probably the only criticism that I think is probably fair. I must admit that Bradley has been pretty inconsistent from game to game, and he seems to be the type to have an 8 point game one night (with 1-5 threes) and then a 27 point game the next night (with 4-6 threes).
That said, NBA players on the whole aren't as consistent as most people probably think - not unless you're talking about bonafide All-Stars / superstars like Kevin Durant, Lebron, Bosh, etc.
Size
I don't think it's fair to bash Bradley on his size because
1) His long wingspan and outstanding athleticism largely make up for a lot of his lack of height
2) The general lack of size in the NBA these days as it is - there are a ton of SG's out there these days between 6'1" - 6'4" these days, and teams with small guards (e.g. Phoenix) don't really seem to suffer much for it.
Now you can argue that the other's mostly have PG skills, while Bradley does not. True, but I personally think this is somewhat irrelevant because the fact is that much of the time Bradley is on the floor he's matched up against guys who are within 1" of his height. The only time Bradley's lack of PG skills ever become a legit concern is if he's put in a situation where he is actually depended on to play the PG spot, which on this team (with Rondo, Smart, Pressey and Turner) is going to happen pretty much never.