The key to recognise is that they're not only storing the images in the web browser, they're also doing all of the model training and inference in the web browser too.
All of it is being done on their own computer.
When you say that they saw ML4Kids hanging, what this means is that *their web browser* was hanging because the amount of work needed exceeded what *their machine* was comfortably capable of. That's a factor of the CPU and memory available on their physical computer, and to a lesser extent a factor of whatever other applications they were currently running.
It's difficult for me to make broad declarations about sensible maximums - for example, I suspect I would've been able to train the sort of model you describe without problems, but then I'm fortunate enough to probably have a faster and more powerful computer than you're able to provide to students.
If you want absolute lowest-common-denominator recommendation, the 100-images limit that I normally use (albeit influenced by cloud storage) is easily trainable on even low-powered devices such as Raspberry Pis.
Kind regards
D