Which effectively pins a different key in the binary. Might as well use certificate pinning, which provides both signing and encryption without a second layer of crypto.
As a sibling comment has also pointed out, signing the data won't help against a user who's modifying the client. You can change the signature the client is expecting on your certificate... and you can also change the signature the client is expecting on your data.
So cert pinning in this case would go from "anyone with a pihole or equivalent can block YT ads on their Apple TV" to "people who jailbreak their Apple TV and install a cert pinning breaker and have a pihole type setup can block ads".
If you are google and your goal is to get people to watch ads, cert pinning is clearly a win if you are at all worried about things like the OP. Clearly they are not very worried, presumably because the bar set by the OP is already too technical for most people (even if it was packaged in a more consumable way).
From a user perspective, you'd want to break all TLS connections and sniff every bit of data on your network, but that's not the perspective Google has when developing their code.
Signing data is just duplicating TLS security measures with a second key. TLS already signs the data, Google just needs to verify the root of trust.
You might be interested to know that this is official policy on Android. Yes, I'm appalled too.