Tiny houses are really pretty awesome, and I think they can be a great solution in a variety of situations. I spent about 3 months aggressively researching a "tiny house on wheels" as an alternative to our current home (2,000-sq-ft townhouse). You are right that there are financial savings, especially if you live in a high cost-of-living area - less energy cost, no mortgage / no interest, no insurance, minimal tax, etc. While my wife and I (no kids) were convinced we could definitely live in a space that size, we concluded there were too many peripheral reductions in quality of life, such that it wasn't worth it, given the ROI timeframe.
For example, if you build a tiny house on wheels, then that means there is likely no legal area to park it inside a city limit (other than an RV park). In most municipalities, temporary housing (RVs, etc) can only be dwelling units for fixed time periods (30 days, etc). So that means you'll either need to park illegally in a city or legally in an outer county area. In my case, if I live in an outer county area, then I can no longer bike to work, which I greatly enjoy, and our driving commute would be pretty bad.
The other big deal-breaker for us was the reality that any place you park will likely have drawbacks and be "temporary." When you buy a house, you generally have a large pool of possible locations. Relatively speaking, there are very few places to park a tiny house. And then, any place you find will only last (in all likelihood) for a few years, at most, and then you'll need to find a new place to park.
All that to say, after analyzing the cost savings and weighing the tradeoffs, it became clear that I'd be happier working 2-3 years longer in our current house, with our current enjoyable life, and then eventually build a small home in an ideal location. Of course, YMMV :)
I would encourage you to really analyze the cost savings. You may be able to achieve similar savings by buying a cheaper place with a 15-year loan (cutting your monthly interest cost, tax, etc). What are your mortgage expenses? How much is principal vs interest vs escrow vs PMI? Let's assume you have $300 / month in principal, in that case it will take you 2.5 years to recoup your $31k in building costs ($1625 - $300 = $1325 current cost - $300 new cost = $1025 monthly savings * 12 = $12,300 yearly savings x 2.5 years = $31k). If your current house will experience any level of appreciation, then your return will take longer. If your tiny house "rent" rises, then your return will take longer.
That said, if I had an enjoyable place to park a tiny house (legally) for the long-term and it was within biking distance of my job and close to the things that mattered, I would likely do it as well for a few years and rent out our townhouse. Good luck!