What role does it play then if it isn't symbolic? Other than the guy wearing the armband can supposedly talk to the referee (doesn't seem to stop the others). It's interesting you should mention Beckham because he is an example of the kind of captain about whom everyone said "him? really? doesn't seem like captain material". He never did the screaming and shouting thing, he just quietly got about his business and led by example, in much the way Lallana does for us. As long as he's not going to be upset about other senior players taking a leadership role as well, I can think of no better candidate to represent the club.
The club captain used to represent the club , you need a leader on the pitch , Lallana is not that sort of person , its no fault he brings other qualities and theres no shame in that .
Stick with Lallana for now. I think a change would be another blow to what must already be a pretty fragile team spirit. I do agree in the long-term though that Davis might well get a look in.