Also lots of people have convinced themselves of lots of silly things. You can probably find dozens of papers from people with bachelors in math (or no degree) claiming they have solved P=NP. A lot of these turn out to be completely bogus, but the authors nonetheless thought they were serious attempts.
Again, I agree doing it at the university is more effective way of doing it, and if you have the possibility to do it, good for you! But most of us can afford to dedicate at most 5 years to studying full time, and than other responsibilities kick in and you can't do it anymore. The majority of your life all the new knowledge you get will come from self-study. So you better learn to do it.
For me, merely pushing myself through a theoretical CS curriculum made me see (and write) hundreds of proofs, hear them explained by professors, and see non-trivial exercises solved during recitations. I don't think you can get the same kind of experience by just reading a textbook, even if it does offer full solutions to problems.
Maybe when there will be full video lectures for both lectures and recitations for the basic math (or theoretical CS) curriculum you could self-study by watching those and solving problem sets. Right now, the math courses offered by Coursera don't seem to match college level, and their platform doesn't really work for proof-based courses like Analysis, Linear Algebra (not the applied kind), etc...
MIT OCW has excellent courses for discrete math, calculus and linear algebra and in some cases videos from the recitations are included.
http://ocw.mit.edu/courses/mathematics/18-01sc-single-variab...
I also stressed a few times already I don't think it is "the same kind of experience". But as long as you make an effort you will make progress and not everyone can manage to fit a university degree in their schedule.
Halmos, Rudin, and Spivak 'Calculus on Manifulds' were at least at one time the main references in Harvard's Math 55 as at
http://www.american.com/archive/2008/march-april-magazine-contents/why-can2019t-a-woman-be-more-like-a-man/?searchterm=Sommers
Your last sentence is on the center of the target of reality.
http://www.cs.berkeley.edu/~oholtz/teaching.html
has homeworks for Rudins "Principles of Mathematical Analysis", and Halmos "Finite-Dimensional Vector Spaces". Finally most problem books (again Springer has a nice selection) have very detailed solutions. Yes, there are proofs that can be done in a number of different ways, but in my experience diverging too far is not very common and in most cases some core ingredients have to make it in in the end anyway. It's impossible to go through a lot of exercises "writing complete nonsense solutions" like that, when you are precisely checking your solutions against the given ones. For many types of exercises there are also simple ways of validating your solution, for example, in probability theory you can often do a computer simulation. In the end that's what anyway has to be done in real world and in research work.