Will apex make a team death match game mode

That's a great idea. 

We've seen it in other games like combat arms, for example, that different modes are even helpful to divide the players giving you a totally different game experience: what i have seen in ca is that average players use to play tdm or fast respawn modes while the slower modes were for player who wanted to play a bit more seriously, listening to the footsteps, givin the tactic its importance and working with and for the team! 

SBMM will just kill that game mode too.

Not unless they just forget match making and make it persistent lobby servers you can join and leave. Then throw in team balance, so player count on both sides even on new joins.

Battlefield 1 even had map vote, so it encourage players to stay on same server and more persistent player count. They never brought this map vote back for Battelfield 5.

Or they could overkill and make it persistent and open world like Planetside 2. (Most epic player count FPS ever). Reminded now again of it, maybe I'll give this another go as its free still.