**THIS IS A HOT TAKE**, which means it's an unpopular opinion.
You probably will not agree with me. Please keep it civil in the replies. If you can't, then don't read/reply.
You shouldn't get Blue Essence or XP for playing Teamfight Tactics because this gamemode doesn't have anything you can buy with BE.
Normally when playing regular League, whether it's Normals, Ranked, ARAM or Treeline, you are playing a champion. You buy new champions with Blue Essence. What else do you buy with BE? Pretty much nothing except for Rune Pages, name changes and the Essence Emporium. And occasionally 1 BE for a new icon when Riot feels generous. None of these things have a gameplay-wise effect on TFT: You can choose whatever champions you roll as long as your in-game currency allows it, your name and icon are account-related and the only thing you can use from the Essence Emporium is the emotes.
So Blue Essence is essentially useless in TFT and those who only play TFT don't do anything with it. Now let's look at what you CAN get from playing TFT. You have event tokens (which, again, doesn't provide you with anything you can use in TFT), first win of the day rewards (more BE and XP leads to more BE also, but i'll touch on XP later) and the newly introduced Beta Pass points. As of now the Beta Pass grants you one Little Legend, icons, arena skins and emotes. All of these are usable in TFT (except for icons, kinda). So why are we crying about not getting BE, which you can't even use in the gamemode you play?
I think in that aspect, Riot is totally okay with not giving BE/XP for TFT. I can understand XP in the sense that playing TFT should still level up your account, as this is a LoL gamemode tied to your LoL account. But as for actual rewards, they should invest in more TFT-specific rewards which you can actually use. As of now, if you have completed every Beta Pass mission and collected every Orb of Enlightenment, you have one Little Legend. Considering the RNG it takes for you to get three or even nine of a non-standard LL with perfect rolls, they could give you more eggs to hatch for free. But also, this is Riot, so that probably won't happen.
In conclusion: BE shouldn't be a reward for playing TFT. In-game (TFT) cosmetics should be, and you should get more than what we're getting now.
Have a good day!