I've had the opportunity of playing one of those $50-$100 case and bow included violins. Now make no mistake: a cheap guitar with a soon-warped neck becomes completely unplayable because the frets will stop working (you press second fret and get fifth with a snare instead). A violin or cello does not have frets and fretboards are separate parts. Nevertheless, the playability was awful. This starts at the strength to need for fingering and ends with the sound quality because there is a difference on working with high-density wood manually following its grain or CNC-milling something out of a block of cheap wood.
Getting a nice tone out of the instrument is the major challenge on bowed string instruments for a long time. Starting on an instrument that would challenge a professional is cheaper but you'll reach a point of diminuishing returns over the expected length of career.
The advantage over a rented instrument is that if your child throws in the towel after all, the instrument stays around in case of a change of mind for trying things.
Obviously you don't need a Stradivarius, but the least you should aim for is an instrument that would be fun enough for a fiddler.
Note that even where no tone wood or other largely manual part construction is involved, like with accordions or many woodwinds, Chinese brands and/or makes without Western companies involved in the manufacture and quality control tend to be comparatively bad for whatever reason.
A few hundred years ago, cheap string instruments were mass-manufactured in the area in the Saxonian/Bohemian border region, with a similar low evaluation of manual labor over machines than might be the case now in China.
Those instruments, when kept in reasonable playing shape, tend to be tolerably good these days: probably there was more of an accumulation of knowledge and competition among workers for manual skills and more long-term perspective than there is these days in China. And when push comes to shove, they had to work for adult semi-professionals with modest means even then. This is not the case for today's cheap instruments. I don't really know the intended target for the kind of instrument I was able to test a decade ago or so.
Basically, it's like training for running on a mud slide: part of your skill set and work is invested in stuff that is only marginally related to future progress. Can be fine for testing determination, but you need to find the right point of time to move on. And of course, you need an active player to figure out when the instrument has already moved beyond playability which can happen in few years (possibly even of just storage) with instruments milled from cheap wood.