Can I reasonably move universities every few years for professional work?

As a data scientist myself, I have found it rather difficult to be a journeyman data scientist in academia. Yes, people need their data analysed; no, they do not want to pay for your full salary, benefits, and office space. Nor are they necessarily going to hire an outsider to analyse their data. They can get their postdocs/grad students to do it.

I have met repeated resistance to being taken onto projects as an outside data analyst. Certain STEM disciplines are especially bad with this. Because I have a PhD, I can find full time positions that allow me to be housed in a department, teach classes in that department, and then take on outside projects as I find them. These outside projects are never enough in and of themselves to provide enough work for me to retain my job. The fact that I can do independent research and teach classes is what allows me to do what I do.

I just think being a journeyman data analyst in academia can be a rather tough life, and you may not have much freedom to actually decide what type of subjects you are exploring. In my experience, you may rank below even the master's students and will possibly be paid subsistence level pay.

I think the other answers have mis-understood the OP's goal. He explicitly says that he wants to be a scientific programmer assisting other folks' research programs, not pursuing his own. Such positions do exist at every major research university. How common they are varies by subfield. In my experience physics departments generally leave all the programming to their students and post-docs, while life sciences department will hire quite a few software developers and data analysts, just as they hire a lot of wet-lab technicians. There is a lot of computational work to be done these days that requires fairly mature engineering skills, but is not novel research. Having your chemistry Ph.D. student set up a secure and robust web application is probably not a good use of their time.

To respond to the OPs question: what you want is not impossible, but you'll have to be lucky to pull it off. While there are scientific programmer positions at every major research university, there are not a lot of them compared to the non-academic world, and they generally don't pay nearly as well. I think PIs find hiring for these positions difficult and time-consuming, so they prefer folks looking for long term positions. Sometimes, one of the functions of these positions is to provide some overlap in institutional memory. Bit rot sets in rapidly each time a student graduates. If you aren't even going to stick around as long as your average Ph.D. student, your utility to the PI is diminished.

On the other hand, many of the grants that fund these positions are time limited. Projects that only last two to three years seem to be pretty common. That would seem to match pretty well with your goals. Unfortunately, at least in large departments, there seems to be a small cadre of programmers that get shuffled around projects as funding comes and goes. You'll be competing with those folks, and as they have a known local track record, it puts you at a disadvantage. For the same reason, constantly switching fields, would also put you at a disadvantage. Software skills are highly portable, but there is always some domain knowledge that's crucial as well.

I think your chances of success will be better if you stick to one field, say the life sciences. It will also help if, early on, you can find work on a high profile project, with a prominent PI.

I'm pretty sure that the funny looks are because people are making the assumption, probably warranted, that you will be fairly shallow in the work you do. Normally academic research is narrow and deep, so this plan seems too unlike what they expect. Some fields take a long time to learn and even more time to be able to deal effectively at the frontier of knowledge. Someone who wants to do a lot of relatively simple (in their view) things has less value, by far, then another who is a well known expert in a specialized field.

Along with this worry is that you wouldn't be a lot of help in leading students to a research (narrow, deep) career as you are a sort of "dabbler" in things.

I don't criticize your plan. Being a polymath (even a shallow one) can be a rewarding life. But deep and narrow is what is more valued in academia.

At non-research institutions (primarily small undergraduate colleges) you might find a better reception. But even there, not having a doctorate is usually a deal-breaker.