Good that you are thinking about your future, but your choice of major should be based on YOUR desire and capability. We once had a new hire, graduated from UC Berkeley with a 4.0 GPA in EE, but they actually didn't have the desire, while they definitely had the capability, and they quit after a couple years to go work in the family's restaurant business. And they likely majored in EE because that's what their family pushed them to do.
On a different note, I knew someone who REALLY desired to be an engineer, but they just couldn't handle the math required.
So, your choice of major needs to be firmly based on desire and capability; if you have both, then finding and getting a job, barring macro-economic issues, should be relatively easy, although at that point, you need to be able to have the soft skills of marketing yourself and being a decent communicator. Whatever major you pick, having relevant and meaningful internships will outweigh general job experience, i.e., getting a job at McDonalds isn't likely to score points if you are trying to be an EE or ME, although if you can fix those ice cream machines at McDonalds, that might be good selling point.
You are currently too far from a post-college job to be worried about the job market, since literally anything can happen to any discipline in the next 5 years. I also agree that some people that can't find jobs are often stymied by their own delusions of grandeur and inability to sell themselves. There's still a sizable contribution from luck and good fortune, insofar as where you wind up and how successful you can be, but you have to have the capability and bonafides to allow the "luck" to come to you. High quality/relevant internships and your performane therein could make a huge difference in how a potential employer ranks you against your competitors.
I would tend to agree that niche disciplines and higher degrees have the potential to limit your horizons to some degree, because an employer has to have enough work to justify a niche position and enough work to keep that person employed and interested in staying in that position. I've personally gone in the other direction, ala being more of a generalist; there's almost always a need for a multidisciplinary role, bridging and liaising between EEs and MEs, say. When business is down, companies will tend to want the flexibility to shift people around, so being multifunctional has the advantage that a company might have not have enough work to employ two engineers, but might hire a single engineer that could do most of both jobs to make it worth their while to hire that single engineer.