All these words are obviously based on 心, although they use different kanji today. You have to keep in mind that kanji is not natively Japanese, but rather a but rather a bunch of Chinese characters representing Chinese words with different semantic scopes than their Japanese counterparts. In the early days of Japanese writing, mostly everything that wasn't poetry was written in a style called kanbun, where it was practically translated into Chinese, and could be translated back into Japanese when reading.
You can read more about it in this thread:
Nuances between the different kanji spellings of あける:明ける vs. 開ける vs. 空ける
What happened with 快い et. al., is not different than what happened with many other words, such has 湖. I'll use 湖 as an example, since it's not abstract at all, and thus it's much easier to explain. ;)
In Old Japanese (and still today), the word for lake was just a compound of the word [水]{みず}, water (and more specifically fresh water, since salt water is [潮]{しお}), and [海]{うみ}, sea. Chinese, on the other hand, had an entirely unrelated words for "water", "sea" and "lake", and when translating to Chinese (for kanbun), Japanese writers had to use the proper Chinese character 湖 - they couldn't make a compound like 水海, since it didn't mean anything in Chinese. Later, the character 湖 stuck as the accepted character for みずうみ, although it was clear to anyone that this word comes from 水 + 海. For the very same reason, salt (塩) and salt water (潮) ended up being written with two different kanji although they used to be the same word.