There do not seem to be any obvious drawbacks. Could it actually be that we discovered a new type of material for building things or clothing that had been overlooked so far?
The mycelium (root structure) is killed in the process. It can then be tanned as you would tan normal leather. MycoWorks is working with Hermes which tans the leather in their own tanneries.
Animal leather is also embossed actually and they emboss the mycelium leather the same way. It’s an incredible fabric that looks and feels like animal leather, but it can also be grown thicker and to shape. It’s very versatile. An incredible product.
Tanning, historically, is quite a chemical-rich and polluting process. Is the tanning process similar for myco leather?
Real, "full grain" leather (the only kind that includes the external surface layer of the hide, is rather smooth. You have to look pretty closely to see the texture, though you will see some imperfections like scars if they aren't sanded away in processing.
Most of the pebble like texture or wrinkled texture you see on leather products is stamped.
This has an interesting set of tanning processes, but it's funny how SEO saturated this topic is. There are some video results that are probably useful, but 99% of my searches are turning up commercial garbage, and nothing to do with actually making mushroom leather.
Iirc from a podcast, sodium carbonate is used to soak amadou mushrooms for a felt-like material, which Paul Stamets uses for his hat.
You'll probably need to learn the jargon and terms of art specifying the exact types of mushroom fabrics and materials that can be produced.
Good luck!