Sounds like your real question is "people are telling me to get into stocks, but I don't want to, should I do it?" - obviously the answer is no. You should not risk your money on something you have no interest in.
However, I also get the impression that you're not disinterested in finance, you are just very unfamiliar with it and think you will be disinterested. There's not much risk in just doing some reading and trying to learn a few basics to see if you are interested. And in saying:
They claim that I'm a very logical and quantitatively adept thinker.
Your family members make a good case that you might find it interesting. Just because you're logical and good with numbers doesn't mean that you will succeed in finance, of course (there have been plenty of math geniuses that ended up doing very poorly), but it helps!
I'm very skeptical of the claim that my quantitative capacity will be of any use to me in investing in the stock market, unless I devote an inordinate amount of time.
Your actual specialized skills probably won't transfer at all. However, being familiar with basic math concepts, as well as with tools used to crunch numbers and analyze data, makes it a lot easier to learn.
Time-wise, there are definitely people who invest effectively despite having a job. It all depends on your goals and specific strategy. Setting up an automated order to buy $1000 worth of index fund shares every month requires barely any time (or skill, beyond the skill of leaving it alone) and is actually an effective strategy - if you can actually stick to it through an inevitable market crash and subsequent recovery.
As for the more general questions:
Can a casual investor effectively predict the direction of the stock market?
Good thing you said "effectively" because no we can easily answer "no". The index usually goes up, so people usually predict that it will keep going up, so it seems like they can predict it. However, every so often it starts going down rapidly, at which point the casuals are likely to panic and sell at the bottom, which results in a big loss.
In other words, even if you just always predict the market will go up, and invest based on that, you will most likely get a good return. However, "always predicting up" is hard to do: During a crash, many people find themselves thinking "maybe this time it won't go up after all" which can make all the difference.
Is the stock market an efficient market, with assets whose direction of change in prices cannot be predicted by casual investors?
The market is pretty efficient, but it is also very irrational. Overall it will tend to favor companies with solid fundamentals and successful businesses. But on any given day you can find many apparently great companies with terrible stock performance, and vice versa. Whether you are wrong about them being great, or the investors, and when (if) the investors will change their mind, are big questions that will impact any decision to invest.