What a brilliant question!
It all depends on you. If you're not living in a major city and you have strong family roots then it might be a little tricky.
Those client/customers who are willing to pay the BIG BUCKS ideally want somebody locally. So again, it's a trade off. If you're not living in a major city then starting something up yourself would be a little harder.
I guess it all depends on your location. To give you an indication, working as a web designer on my local city would roughly get me 15,000GBP per year, in London that becomes at least double, tripple and in some cases quadrouple that.
Since I don't like London much, and I would not be interested in leaving my home city to chase for more money then this would not be an option for me. That being said I could possible travel to London every so often to chase up client work, but again, this all depends on you and your willingless to travel.
In terms of the learning this should be enjoyable, if it's not then I would side towards doing something else more enjoyable. I would not learn something in the hope of finding a job and a career if it's not enjoyable. Our world has changed so much over the past 10 years, particularly on the web which means that this learning process would likely be done again and again and again!.
Just to highlight, when I first coded a website I used HTML4 and I hate to say it Macromedia Flash (as it was known then). Now many web designer and developers have dropped Flash for jQuery and we now use HTML5.
To sum things up, if you stop learning particularly in web design then you'll quickly become old and outdated gradually loosing your stronghold over your field.
The competition depends on your country. Where I live 95% of businesses (small and big) would not be interested in paying more than 350GBP, many of those will even look online for cheaper solution (e.g. 1&1, Google web pages and VistaPrint, outsourced solutions), and this is becoming more of a problem. This is the sad truth, and changing their mentality is a very hard challenge.
This however does not stop me from keeping this as a hobby, I really do love web design, but that being said I don't particularly enjoy it as a full time career option. Mainly because the money is not stable and in many cases, as you've already pointed out, you don't get paid enough for the work involved.
On a similar note, I use to work at Gloria Jeans Coffees, and I was making more money there as a barman than I was working as a web designer for the top firm in the country. Sad, but true.
There are high paid web design jobs, but not only do you have to be in a major city, but you also have to endure at least a few years of "less than a barmade" salary. Is this fair? Hardly.
Unfortunately the market has become flooded with college and university graduates allowing companies to pay people who start a career what they like. This is not just isolated to web design, but pretty much any educated field it today's world.
The only added downside to web design is that it's a little like art, people do it for the passion and not always for a career option, this also changes the outlook of how people deal with you.
Let me explain, if you're interested in freelancing take a look at this https://www.youtube.com/watch?feature=player_embedded&v=zpzGo3jAW_g.
I laugh now, but I have witness all of those kind of clients.
With client work you have to be careful that the customers take you and themselves seriously, and follows up with payment. It's shocking on the amount of people I meet who somehow feel that I should work for free because I enjoy what I do. If you let them you will eventually become warn out.
I enjoy and love web design, so for me it's a very good hobby I would like to keep and evolve it to whatever it branches off to, whether it be internet marketing or just a hobby site which earns me an income.
Hopefully my outlook and experiences have answered your question.