As a side note, 広辞苑 states:
かい 【詮・甲斐】 : 行動の結果としてのききめ。効果。また、してみるだけの値打ち。
I believe your assumption
Assuming 期待できる is used to express a strong possibility or hopeful feeling and だけの + 値うち is used to gauge the worth of the attached rel. clause
is correct, but the overall interpretation is slightly different. Using the definition you quoted, "生きている甲斐がない" can be interpreted as "生きていることに、期待できるだけの価値がない"(*1), which can be roughly translated into "Nothing valuable is going to happen in my life". If I allow myself to be informal due to my not being fluent, it's like "Yes, I'm alive. What can I expect from it because I'm alive? (There should be something, right?) Nothing, nothing truly valuable!"
新和英大辞典, a good Japanese-English dictionary from 研究社, has
Life [is meaningless, is not worth living, has no point]
for "生きる甲斐がない". I think these are quite good translations.
So yes, this phrase is used by people who are unhappy, and who don't expect to be happy in the future.
(Careful : anything below is my subjective point of view)
It is important though, that the speaker is unhappy because of the lack of some essential value in his/her life, and not because of excessive amount of sadness. This phrase is sometimes used with "without you/her", to express how deeply the speaker cares about him/her and how crucial his/her existence is, to the point it affects the meaning of his/her whole life.
(*1) : This phrase is can be a little confusing --- without "、", this can also be interpreted as "「生きていることに期待できる」だけの価値がない", "there is no value good enough for me to expect a good deal from my life", which does not really make sense without context. I used the phrase as "生きていることに、「期待出来るだけの価値」がない" here.