The issue with him in possession isn't that he isn't solid. He's very secure, 84th percentile among fullbacks for basic pass completion last season. His pass completion compared to our other fullbacks
AWB - 84%
Shaw - 83.7%
Dalot - 79.2%
The problem is how he has to play to remain that secure.
Here's how those three players' attempted passes break down between short (5-15 yards), medium (15-30 yards) and long (30+ yards) last season, percentages rounded.
AWB - 63% short, 31% medium, 6% long.
Shaw - 52% short, 37% medium, 10% long.
Dalot - 38% short, 46% medium, 15% long.
In other words he offers a much more limited range of passing, with those medium/long pass attempts being 80th/72nd percentile for Shaw, 84th/85th percentile for Dalot but just 20th/4th percentile for AWB.
That conservatism is reflected across other stats too, with AWB being 30th percentile for passes into the penalty area, 58th percentile for passes into the final third, 46th percentile for progressive passes, 33rd percentile for progressive passing distance, etc. Both our other fullbacks beat him across all those areas, with his rival Dalot being 77th, 67th, 58th and 68th percentile in those stats respectively and Shaw generally better again.
Outside of passing, while AWB is better than Dalot in terms of progressive carries and attempted take-ons, Dalot is statistically more likely to carry the the ball into more advanced positions (final third/penalty area), not to miscontrol the ball and not to be dispossessed.
All of which is a whole lot of numbers just to say that AWB is limited on the ball and, presumably because he knows this, plays within those limitations. Which is very understandable and I don't think those limitations should see Dalot start ahead of him at the moment, as AWB has generally been in better form recently.
But looking beyond this season a fullback that limited and ineffective in possession really isn't good enough for a side looking to win major trophies. He gets the ball in space quite a lot as teams much prefer to see him in possession than Shaw on the other side, and we really need our fullback to be able to do more with that possession than predominantly play square to his CB or short to the RW in front of him who so often has to go backwards from there anyway.
The difference between our RBs and Shaw (who keeps the ball well and does effective things with it) highlights why this is an area that needs fixing next summer.