You ask if there are serious issues as to why it might not work and I'm going to tackle that (I might point out that I did travel for my job over 50% of the time for 9 years and know what a toll it takes on your life):
Flying in every week means basically giving up weekends. You need to fly out on Sunday night to be there for Monday morning and fly home Friday night after work hours or Saturday mornings. You are always packing and unpacking. You would end up using some of your leave when your flight gets cancelled. You are going to be there 5 days out of seven and travelling at least part of the two days you aren't there. So really you are living there. It makes more sense to move, get the experience from the interesting job and then move to a different place later if you don't permanaently want to stay there. Plus without transportation, the things you can do outside of work hours are severely limited.
You say you would take a lower salary to account for the cost, but have you truly calculated out how much a week it would cost the company? It's not just the airfare and the travel expenses but also the administrative costs to manage this. That's a pretty big hit to your salary.
It sounds as if the jobs you are looking at are in large city since you assume mass transit. In that case they probably have far more local applicants than jobs. Usually when that is the case, the person from outside the area would have to bring something pretty special to the table for a company to consider such a thing. Ask yourself if the skillset you have is that interesting to a company.
And from a company perspective, I would be far more likely to want to hire someone from another area to telecommute than pay their airfare every week. The person would be more likely to be happy and be more likely to want to stay employed.
If you are single, this schedule makes it difficult to be in a long-term relationship. If you are married or currently in a long-term relationship, this will put a great deal of stress on it. If you have children, this schedule is utterly unfair to the spouse who stays with the children and is unfair to the children.
It is difficult to pursue many hobbies when you have a travel schedule like this or have pets. So really your life ends up being work and sleep, do you really want to commit to that?
If you have a family emergency, you will have to pay out of your own pocket to return to your home city if that is where your family is. If the reason why you don't want to move is your family, then consider how much less time you will have available for them and how much harder it will be to be with them if something happens. I know I got a lot of family resentment because I could not make it to some family funerals when I was travelling. This is true if you move too, but people are more likely to consider this when moving than when travelling.
Weekly travel is hard on the body. It also makes it hard to work the weekend as is often needed in the software industry.