I think I've noticed something which partially explains why not all servers are affected by the bug, it's kind of obvious in hindsight.
Regular players will know that, for whatever reason, sometimes on some server platoon tags aren't visible; it's been happening intermittently for years now. As it turns out, when a server can't display platoon tags it also doesn't make the clients stutter! It's like having Player-Created Content disabled server-side.
This to me suggests that if EA wanted they could implement a workaround (or even actually fix the bug) server-side, without going through the trouble of patching the client, and possibly without even touching the game-server code either, if the problem is entirely on the back-end.
Footnotes: I'm focusing on platoon tags because they're the easiest to notice (just open the scoreboard and see), so far I haven't been able to observe if this also affects player emblems. Also during my testing I found one server where, by chance, nobody was representing a platoon (confirmed through the player profile API) and yet the server still caused stutters, presumably because the Player-Created Content delivery system still had to look up players and thus stumbled into the bug.