I want to get in front of this first. There are tons of jobs which absolutely require a post-secondary education. Sometimes this is legally required, but usually it isn't, and unless you have some exposure to the field you might not even understand why.
Skilled trade apprenticeships are post-secondary education. Electricians and machinists spend half of their apprenticeships inside a classroom, according to the journeymen I know. I don't know about other trades but I imagine it is similar. I know this isn't a university degree, but I wanted to point this out because there is a very big difference between "jobs that don't require post-secondary" and "jobs that don't require university degrees". (This extends to most other vocational programs, which are either legally required or are obviously beneficial training for employers in need of those skills.)
Professional schools require all or part of a university degree, which varies from school to school. Accounting schools typically prefer a mathematics, business, or finance undergraduate, but some will take an undergraduate transfer once they've completed enough math credits (per accountant I know). Nursing programs usually take undergraduate transfers. Actuarial sciences require a completed undergraduate mathematics degree. Law schools require a completed undergraduate degree, usually from the humanities (proof you understand how to argue effectively). Medical, dental, veterinary, and pharmaceutical programs all require a completed undergraduate degree, usually in the biological sciences. There's no reason you specifically need to have a degree, but you'd need to do most of a degree in order to satisfy the prerequisites anyway. Professional school is usually legally required to practice those professions, so if an undergraduate degree is required to enroll in the professional school, effectively the profession legally requires an undergraduate degree.
Engineers are also legally required to complete an undergraduate degree.
Then you have all of the jobs that don't legally require a university degree, but really do in practice. That includes every kind of research job, whether working for academia or industry. Research assistants really must have an undergraduate degree in a relevant field, otherwise they won't be effective assistants. Researchers must have a doctorate in a relevant field, otherwise they won't understand how to identify, evaluate, or document an original innovation (this is the purpose of a PhD).
Then you have all of the jobs that don't technically require a university degree, but require knowledge that you are only likely to obtain while completing one. This includes any job that requires any sort of quantitative analysis, because the necessary skills are not taught anywhere at a high school level. Due to the depth, breadth, and abstractness of the necessary skills I also do not believe they ever could be taught at the high school level (in particular, the prefrontal cortex - responsible for abstract thought - isn't fully developed until you're 25). Financial firms hire quant 'generalists', so they snap up anybody with a physics, statistics, mathematics, or computer science undergraduate. Other firms require quant specialists, who need to have additional specialized knowledge you aren't going to see at a high school level (more about this later).
Then you have all of the jobs that don't at all require a university degree, but which employers have good reason to prefer. BFA or MFA for design and illustration jobs shows you have practice, peer approval, and experience dispassionately reviewing your work. Psychology for HR. Psych, sociology, BComm, marketing for marketing and advertising, which at least shows you understand the trade language (and sometimes means you're a quant specialist). And speaking of language, a linguistics degree or an undergraduate language degree makes it a lot easier to hire you if you're a translator.
So what are the jobs that don't require a degree? Non-professional jobs with no legal degree requirement, that don't involve original innovation in a particular field, don't require knowledge you can only obtain in an undergraduate program (particularly don't involve any sort of quantitative analysis), and don't involve any tasks for which you performance improves with a degree.
Help me if I missed any, but from what I can tell this leaves exactly:
- Unskilled manual labor
- Retail and customer service
- Light clerical work
- Management (iff you're managing the above)
- Skilled trades (still post-secondary, just not a degree)
The last destroys your body. All of the skilled tradesmen I know are looking for ways out and they're only in their 30s. But at least it's good money as long as the cartilage in your knees holds out.
From personal experience I can tell you that the first four are ****ing horrible jobs. I'm glad I lived in a country with relatively cheap tuition, but even if I didn't there's really no amount of debt I wouldn't have taken on for even a chance to escape them. Those jobs don't pay nearly enough to be worth doing. Ignoring the labor side of things, though, the demand crunch that would happen if kids actually listened to these know-nothing do-nothing billionaires, and chose these four jobs over getting an undergraduate degree, would make our entire society collapse overnight.
Anyhoo... you had a question about computer science. I'll answer it now.
The issue is competitive advantage.
You can definitely get a software job without a computer science degree. Lots of companies need light custom software work, like someone who can write an intranet app to run a database query, or a new Wordpress theme. Mostly jobs where software isn't the core competence of the company. Where the software needs to exist or otherwise improves efficiency, but isn't a major product differentiator or a source of competitive advantage for the firm.
A red flag for me is when a software company hires a lot of people without degrees. Companies where the software is the product, or a major differentiator, or a major source of efficiency for the company.
The thing people don't really understand is, a computer science degree doesn't teach you how to write software. It doesn't even try to. Computer programming and software development are works of art and creativity, and there's nothing in a computer science degree that can help you achieve a higher art form the way a BFA would - the concern is always correctness, not the artistry. The same goes for the imo purposeless software specializations, like software and computer engineering.
What computer science does teach you is that quant stuff, which you pointed out. Quantitative analysis is a very important but casually neglected aspect of commercial software development. Algorithm and data structure analysis are only a small part of this toolset. A computer science degree gives you all of the tools you need to model and evaluate every part of your software system, including product development and even your engineering processes. In effect, computer science graduates are quant specialists in every part of software design, project management, marketing, operations, and support.
Software companies that don't discriminate between software developers with and without CS degrees probably don't see the benefits of an educated engineering workforce. That either means they aren't working on problems hard enough to need systems analysis, or they simply don't allow their software developers (read: code monkeys) to participate in the company on the productivity improvement level. As a software developer myself, it's a red flag because I know I'll never get to do my best work at that company. As a customer and potential investor, it's a red flag because I know this company has no competitive advantage. Their 'moat' is hours worked, and that's not good enough. That's the kind of company that will get disrupted by a lone genius working in their garage.
The up-side for non-degree-havers is that you'll be the last person to get laid off, I guess.