Shea Weber's generally been a bit overrated, doesn't mean he wasn't a top tier-ish type of D in the league for some years, but he wasn't this all-time two-way beast either. Firstly, his defensive game (while solid) got a bit overrated, probably because he was a big physical scary-looking guy with a Paul Mara type of beard. But this is a player that got continually danced by Ryan Kesler in a playoff series, and Kesler while good was never that much of an offensive virtuoso, but with his speed Weber just couldn't keep up.
Secondly, Weber was a subpar puck transporter and playmaker, two fairly critical aspects of a 1D's game. His highest amount of assists in a season was 33. That's very low for a #1 type of D that logs shit ton of TOI year in and year out. Even his former colleague Ryan Suter, who himself was never that much of an all-time playmaking ace, and played on the exact same type of teams as Weber, either tied or eclipsed that assist amount 10 times over his career.