The result is that your number system (a) makes many common operations dramatically more computationally expensive, (b) has less predictable rounding which is very tricky to reason about, (c) generally gives worse results for the same bit budget. Instead of e.g. evenly spaced numbers, you get spacing like https://en.wikipedia.org/wiki/Minkowski%27s_question-mark_fu...
One thing you can try is storing a floating point numerator and a floating point denominator, and renormalizing them by bit shifts instead of finding GCDs. This lets you avoid rounding errors for small ratios. For general purposes this advantage isn’t really worth doubling the number of bits and complicating arithmetic for though.
See e.g. https://observablehq.com/@jrus/qang
But maybe that just results in all the floating point weirdness again, just not for small rationals.