I think this is a great exercise and can really help you with your photography — even if you end up using zoom lenses in the future. And, there is a commonly-used "trick" here — it's called One Lens, One Camera, One Year, suggested by longtime prime-lens enthusiast Michael Johnston (also see his The Case Against Zooms).

A Method

The original suggestion was specifically to use a Leica for one year, with one type of black-and-white film:

I argued that it would improve one's photographic chops in many ways. It would make you stop thinking about camera and lens options; the use of the one camera you were using would become second nature; the "transparent" nature of the RF viewfinder makes the finder image less seductive, less easy to get lost in (view camera photographers know how easy it is to get enthralled by that gorgeous image on the groundglass); the minimal shutter lag and mechanical responsiveness of the Leica encourages you to learn the benefit of timing the moment of exposure exactly; the necessity of developing film tends to make you more conservative and thoughtful and avoid shooting too much; and learning to see in B&W is a good foundation even for color photographers because color can't substitute for meaning, and value (tones) comprises the structure of many good photographs, even ones in color.

The One Lens, One Camera, One Year concept is a broadening of that; Johnston suggests a 50mm-equivalent prime lens, and has a specific methodology and rules to follow. And, covering the "commonly used" aspect of your question — yes, people have picked up on this. See this post on Petapixel, or just search for the phrase to find many examples.

Scaling it Down

I think you can get plenty of benefit from even a scaled down version — start with one month or one week or one weekend photo-excursion. Instead of shooting with one black-and-white film, try to shoot in one style or with one particular goal for that period. This is like practicing working with sonnets or rondeaus or villanelles instead of free verse — sure, it's restrictive, but that kind of constraint can be a useful tool for creativity.

The key is: when you're "jumping off" the hook, dive in the deep end — just start doing it, and don't leave yourself an easy out.

Personal Note, and, On Travel

As a personal anecdote, I've shot exclusively* with prime lenses for the last ten years, and haven't missed zoom in the least. Of course, I don't do wedding photography or sports. But, you know, none of us really does everything. And I find my primes — or my phone, for that matter — perfectly adequate for travel photography... sure, my results aren't postcards, but the best results in those locations are going to be taken by someone who has taken months to scout and prepare and wait for the right moment.

That doesn't mean that I don't take photographs while traveling — and I have some that I think are real keepers — but it frees me to remember that I don't have to get That Shot of the Taj Mahal or the Eiffel Tower or whatever. That's already been done; I can buy a postcard, and can focus on the kind of photography that suits what I have with me.

Focal Length "Coverage"

A common worry is making sure you have the range of your zoom lenses covered in primes, with lots of steps in the middle. People often do a lot of things like analyze the metadata of all their photos to find most common focal lengths, and etc. My advice is to not bother with this. You only need two or three primes, and you will be surprised how quickly you adjust.

More Personal Anecdata

Currently, I have a 23mm and 56mm for my Fujifilm, which give a field of view about the same as the "traditional" 35mm and 85mm focal lengths on 35mm film — so, basically I have a wide-normal and a portrait lens. Previously, I had a 15mm, 40mm, and 70mm for Pentax (wide, longish-normal, and longish portrait), and when I switched systems I thought I'd have to build up the same in order to keep doing what I was used to, but it turned out to be relatively easy to switch mindsets.

More

Oh, and finally... you may also find How do I compose photos with prime lenses? to be helpful, or at least interesting.

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* except when borrowing a camera or using a point & shoot for a bit or something.