Gen 1 carts are a lot more durable than people give them credit for. The whole reason why Gen 2 died off so fast was only because the clock was constantly running on the battery.
Your Yellow still probably has a save unless it's been kept in terrible conditions ie. under a lot of stuff in extreme temperatures. Out of the all of Gen 1 carts that have been in my possession over the years (about 20, usually just asked to clean them and return them because I'm apparently the only one in town with the right security bit), only one of them is "broken" and it's not due to the save battery, it just crashes after about 5 minutes of playing it (likely a corrosion somewhere). The Japanese Red, Green, and Blue copies I got last year all still had their saves and those were likely made before the west even had the games yet.
I'm actually surprised how much of a beating those carts can take. I once revived a friend's copy of Crystal that was sitting in his garage and had a mysterious black film on the board and contacts. The thing still boots, maybe not the first time but within 3 attempts. Of course, it can't save, didn't really feel like touching it more than cleaning the contacts so I didn't get the battery out.