Ideally they would get washed at the hospital. Hospitals wash massive amounts of linens anyway, I can't imagine the additional expense being material. Mesh bag + name tag costs what, $1?

And if for some reason the clinic doesn't wash linens, well, presumably the scrubs are being washed at home now, so that's always an option. The "special bag" you refer to can be gym bag or a disposable plastic bag. That is still miles less gross than the status quo.

If a gym-goer can be expected to not work out in street clothes/shoes, surely similar standards of hygiene can be expected of healthcare practitioners!