Yes and no, I think. You’re right, it is possible, but you and I started out during a time when companies tended to hire based on portfolios alone with degrees being an added bonus. Today, without that 4-year degree, lots of design applicants (especially those just starting out) are eliminated before they’ve even shown their portfolios. Where I’ve been working for example, no degree means no job. These people will likely never have the chance to learn from those seasoned professionals with art directors and senior designers, like you did. Today, those opportunities (at least in North America) are typically reserved for those with the pre-requisite degree.
You sort of lumped designers and web developers into the same category, which might not be entirely appropriate. Development skills, as far as I’ve noticed, still outweigh a degree. Learn PHP or JSP inside and out, and a job is waiting — with or without the degree. That’s not true for new designers where there’s a whole lot more competition.
As for it being easier to learn on one’s own today as opposed to previous decades, yes, but it’s also a whole lot easier to learn the wrong things. There’s lots more materials out there to learn from — some good and some bad — but a clueless beginner doesn’t know enough about it to know what he or she doesn’t know and needs to learn. Most people study what they’re interested in while neglecting what might be tedious or difficult, which are often the very things they should and would be studying in a formal, well-rounded design program.
In addition, the proliferation of terrible advice on homemade YouTube how-to videos with thousands of views is good evidence that self-learning beginners are focusing on the wrong things — likely because they don’t know the difference.