What is the grammatical reason for 的 at the end of short responses such as 好的, 是的 and 會的?
的: positive response? Not exactly!
The function of 的 is well stated in 漢典:

4 . 助词，用在句末， 表示肯定的语气 ，常与“是”相应：这句话是很对～。
At the end of a sentence, 的 is an auxiliary word, indicating a response with emphasized confidence. It is often used in the &quot;是……的&quot; structure.

I think it's not proper to literally translate 表示肯定的语气 into &quot;indicating a positive response&quot;. That's because, although the example sentence

这句话是很对的。 This sentence is quite right.

is a positive response, a negative response in the 是...的 structure perfectly make sense too!

这句话是不对的。 This sentence is not right.

The 是...的 structure has been well studied in this question. See golddc and 孤影萍踪's answer.
的 in a short answer
Come back to OP's question. Indeed, when 的 is used in a short answer, it's generally a positive answer.

Q: 你會彈鋼琴嗎？ A: 會的。
Q: Can you play piano? A: (Yes), I can.

But I would rather say, this effect is related to psychology. Most of the time, we tend to be confident to give a positive response; however on the contrary, a negative response is seldom stated with confidence.

Q: 你會彈鋼琴嗎？ A: 不會的。

The grammar is perfect in this sentence, but it would sound like:

Q: Can you play piano? A: Of course I can't.

You would feel such an answer sounds cynic.
For a neutral question (I mean, whether a &quot;yes&quot; or a &quot;no&quot; answer won't be specially expected), a negative response with confidence is commonly used too.

Q:他會來嗎？ A：不會的。
Q: Will he come? A: I'm sure he won't.
Q:你是黨員嗎？ A:不是的。
Q: Are you a party member? A: Of course not.

It is grammatically perfect and sounds right.
Conclusion
In Chinese, 肯定的语气 is a little ambiguous. It can mean either &quot;positive mood&quot; or &quot;confident mood&quot;. In this auxiliary word 的 case, 肯定的语气 is better to be considered as the confident mood.