Hell yes, in a heartbeat. I wouldn't even think twice, and I think anyone who wouldn't needs to seriously sit down and have an objective think about their decision.
First of all lets look at what Avery Bradley is right now. On the downside he is relatively undersized at the SG spot, and he lacks the ball handling, court vision or passing skills to be a legit option at PG. On the upside he's averaging 17 Points and 5 rebounds Per 36 minutes, he's a former All-Defensive 2nd team selection (and one of the best perimeter defenders in the league), he has an outstanding attitude (good work ethic, modest, highly coachable, extremely high motor), he's very athletic and his shooting percentages (44% from the field at 35% from three, 75% on free throws) are pretty solid for a guard who has been forced to take a lot more shots than he has previously been used to.
With Bradley the good strongly outweights the bad, but here's the key factor in this decision. If you aren't willing to sign Bradley to 5M/year, then who do you seriously see out there who you think would sign for $5M a year, be a better player than Bradley, and is young enough to fit with the team's current direction?
I'll answer this for you - nobody. There is not a single player on the market who is < 25 years old, who can give you 17 points per 36 minutes with 'decent' shooting efficiency, and who is a who is a well above average defensive player. Even somebody who meets all of those criteia and DOESN'T play defense will still cost you way more than that - guys like OJ Mayo and Eric Gordon are getting pay days in the $8M-$10M/year range and are below average defensive players.
In reality for $5M at the guard position you might pick up a 30+ year old veteran like Jamaal Crawford or Ray Allen who will give you good constribution in one area (scoring) but nothing else.
Otherwise you will get a younger guard who can score and the skills to be a combo guard, but you'll have to take the gamble on a guy with a bad mentality that will lose you as many games as they will win as a result (somebody like Jordan Crawford).
To pay $5M a year for an athletic 22-23 year old guard with a great attitude who can dominate a game on the defensive end of the floor and still give you a solid 15-17 points a night would be one of the greatest bargains this league has seen in the past decade. Lock that guy in for 4 years at that price and you'd be the envy of all the other teams out there based on his potential alone.
Based on Bradley's skill set he is a legit $8M/year player and is well within his rights to expect that type of pay cheque. He is very much the type of player who could turn an average playoff team into a legit contender due to his attitude and his ability to perform on both ends of the court. Wheter Boston would be willing to offer that is another question altogether, but I assure you if they don't offer it somebody else will.