Is it true that jobs don't care what subject a Vet gets a degree in, b/c jobs just want to see that they had the discipline to complete one?

It comes down to what you want to do in a civilian career. If you want to be an Engineer you will need an engineering degree. If your goal is to work in accounting, a finance degree will likely be required. Most larger companies are held to DoL standards and compliance that require what are called Labor Categories (LCATs). A labor category is going to be a required level of education and years of experience and may vary by company.

For example: a GEOINT Analyst I may only require a HS diploma and 3 years of relevant experience. A GEOINT Analyst II may require a HS diploma and 5 years of experience. Meanwhile a Software Engineer I may require a BS degree in Engineering or a related technical field like computer science is required plus 2 or more years related experience

I would encourage to take a look at Job Descriptions for civilian positions that you may want to target when you get out. That will help guide your decisions in regards to higher education.

If you have any questions, feel free to message me directly and I will be happy to provide guidance from a civilian POV.

I worked as a manager in both public and private sector and had a hand in hiring dozens of people, mostly veterans. Here's my view on education: get as much as you can.

In public and private sectors a degree is often an "opener" for job consideration. I mean having a degree appropriate for the level of employment drew me into reading the rest of the resume. Some jobs require a degree or equivalent experience as stated the job description or announcement. Most managers look for the degree first.

In some cases the type of degree matters, but for many jobs it doesn't. You'll see things like "Bachelors Degree in business, accounting, marketing, or similar is required (equivalent experience considered)." In this case, most any business oriented degree will do. Equivalent experience may be spelled out later in the announcement. In other cases you may find, "Degree in information technology, computer science, or automation engineering required with accompanying certification as an Oracle database administrator required." If you see something like this, the company or agency is looking for pretty specific education and certifications. If you don't have them, don't take time to apply.

The school from which your degree was granted almost always doesn't matter. However, it does matter if the school has a reputation for passing out bogus degrees requiring the "students" to do no real work to get the degree, i.e., you pay, you pass. Most HR departments recognize these schools and caution hiring managers about "pseudo degrees." On some occasions you may run into a business that feeds almost exclusively off certain universities. Look at the backgrounds of the senior executives if they are available on line. If almost all of them graduated from the same few universities, e.g., Harvard or Yale, then that's a hint. If your degree is from Mississippi University for Women, you might not bother to apply.

What your leadership is telling you about getting degree work is largely true. Most curriculums have "general education" requirements like math, physical science, life science, history, English, and maybe foreign language. If you take these undergraduate courses on line or from a university on your installation, they probably will transfer to most other schools. Be cautions of taking Junior and Senior level courses or specialized courses if you aren't planning on graduating from that particular school. Many universities also have a minimum number of hours you must complete at their school to get a degree from them. Some may waive this if you are a veteran.

The end degree is really up to you. Don't pursue a degree in a field you don't like or can't excel in. I didn't like math, so I sucked at engineering which I tried for two semesters. I loved history and political science and aced those courses. That's what my degree is in. On installation education offices may have guidance services that will help you target possible career fields in which you would excel or at least enjoy. There's lots of interest and aptitude tests available, but be sure to have a counseling professional help you interpret the results.

That may hold true for the service and any degree would give you a leg up on the competition who didn't have one. My experience in competing for civilian employment is that you need the degree that corresponds to the job you're competing for.
I was an Army Engineer in the service and my associates degree was in Construction Management which fit my MOS. It made me competitive to and through CSM.
To be competitive in the civilian construction management field I had to work through a BS program for Engineering administration which was engineer specific yet generic enough to apply to a wide range of jobs within the career field. I used to joke with my kids and grand children that I wanted to be an engineer when I grew up.
My recommendation would be to figure out what you want to be post military and gear your courses towards that goal.