I found this website but it seems a bit self-contradictory and doesn't seem all that scholarly. That being said, it seems to be suggesting that it more evolved from other ball and wall games played in the Basque region which makes sense. I would guess the cestra evolved later for tha game as the games became more competitive. The website says it was played just by hand, played with gloves and then with the cestra which would make sense. It certainly could be effective as a weapon but I think it might be like a golf club. Sure you can use it as a weapon and it could be a very effective weapon but it wasn't designed for that and so people didnt think to use it like that. It is incredible how fast you can throw balls with them. I think it's telling that the website has a seperate section for "Common Injuries"

https://thesportofjaialai.weebly.com/history.html

Well, yeah, in all websites I’ve researched they only talk about the sport and never say anything about it being used as a tool/weapon, still I would love to find a proper research paper on the topic. Maybe nobody evaluated the topic from this approach because they focused only in the sport part. But I think you are right and not big chances of the “cesta” being broadly used as a weapon. Maybe in some riots.

Here you can see a most hilarious video from a movie from the 50’s, the real craziness begins around minute 5:40 LOL

https://www.youtube.com/watch?v=EZG7VC9QYuQ

in one of the modesty blaise books - willie garvin uses a cesta to throw an improvised grenade.

In another he uses a bra as an improvised sling.

Willie garvin is one of my fictional heroes