Back in the 70's where someone could start out as a drafting assistant and then progress up and finally become an engineer? Many.
Today? At least for engineers in our facility (with over 800 engineers) you MUST have a degree (in engineering, math, computer science, etc.) to be an engineer. There have been a rare few special cases where they have considered on the job experience and bumped someone up, but I would say less than a handful got that privilege. I can't speak for other companies and their policies for engineers, nor for any of the other professions you mentioned (although I believe in our state you need a degree to be a teacher. Otherwise they are an assistant that requires working with a teacher with a degree.)
And no, I don't want everyone to be a trades person, just as I don't believe everyone needs a degree - especially with college loan debt. Not trying to pick, but this is kind of an indication of some of the problems today here. There is too much ALL or NOTHING. There needs to be plumbers, carpenters, electricians, truck drivers, etc in this world. We also need doctors, engineers, teachers, scientists, etc. The secret magic is having that right blend. There is too much push for people to get a degree, and the trades are hurting for people. But that does not mean shut down all the colleges so no one gets a degree so everyone is a trades person.
Free for anyone? - I do believe there should be some sliding scale based on need (should a billionaire's child get a free ride?) and grades/effort. Sandy above indicated how their company reimbursed based on performance (100% for A, 75% for B, etc). Should someone who just wants to stretch out an education and "live off the system" forever get a free ride? I recall when I went there were some "professional students" that slacked off, took minimum credits, and had no interest in actually graduating. They were on their 7th year for a 4 year degree, and not just going part time because they had a real job. I knew another person who had no interest in becoming an engineer, but his parents forced him. He failed every class on purpose so he could tell his parents he "tried", but now "I want to do what I wanted to do in the first place" (which was to be a telephone repairman and not an engineer).
And sorry, I do not agree there is a "societal benefit" for someone to have a degree in 18th century Bulgarian Flute Music (and I sincerely apologize to any flutists, Bulgarians, musicians, or anyone from the 18th century that I may have offended by this random selection). Based on what is happening in today's society, there are many educated people that don't appear to actually use their brain..... Just my personal opinion