There is certainly information to be gleaned from seeing an individuals level, but honestly I think the current information is, if anything, detrimental to the game. As many have said, I think all level really shows (if it wasn’t being exploited) is what tier of experience a player falls into; level 5 and under, they probably have virtually no idea about class roles or team-based objective play. From levels 5 to 10, you can generally expect a reasonable understanding of the game but there is still a lot of variance, and then individuals over level 10 more often than not seem to at least understand the mechanics.
What I frequently see is people getting frustrated and raising a fuss that their opponents team has more higher numbers on the level chart than theirs, and of course the usual bickering about K:D ratios and scores. While honestly I feel that the score calculations by xp earned are relatively fair (certainly there are exceptions, even excluding exploits), K:D is generally not a very valuable statistic for me.
I think it could really get people into playing different roles and tackling objectives more if the score screens showed more statistics. Instead of pushing people to kill more players, why not break down xp earned by role (i.e. support xp, game mode xp, combat xp.) In fact, why not give players separate levels? Perhaps someone could level up as a support player separately from their level earned as a combat player or from capturing objectives. I think the more granular the data available in matches, the better people might feel about their roles in the game, and it could help people see that there is more to do than just spray bullets down the road; if possible, why not show how much xp someone earned from healing, or how much ammunition someone gave out as separate statistics?