I think your issue is just that your grammar was wrong.

是的(Shide) is the closest thing to yes in Mandarin, but it's as universal. Answering 是的 on the question 你会中文吗 is like answering "yes I am" on the question "do you speak English?". 可以 isn't correct either, unless they asked 你可以说中文吗？, which is more like "Can you switch to English?" And not "are you able to speak English?" if I'm not mistaken.

Answering 我会 is perfectly fine, even if you don't speak perfectly Chinese. If you want to sound be more humble you can say（我会）一点点/丢丢. 还可以 doesn't fit the pattern either, but it is ok too I think.

Not a native speaker though.

More on topic: To me not bad/不错，no problem/没事 are just as positive in English, maybe even more in the case of no problem. But I'm not a native speaker of either.