Sounds like this gives the NYC labor standards folks to accept complaints and sanction bad employers. I'm imagine that requiring verbal contracts would be something that you could complain about.
Written agreements don't necessarily mean that you need lawyers involved and are always a good idea unless you're getting paid on demand. When I was in a corporate gig as a second level manager, I could sign off of statements of work under $50k with a contract framework. I'm sure an entity like freelancers union will create a base agreement for these sorts of things, if they haven't already.
Especially the section where his attorney starts talking (at about 7:20).
Usually you want a contract to specify IP rights. You own the work you author, whether you're paid for it or not, unless you specifically transfer ownership of the IP in the contract. If you're a programmer, you may assign the IP rights of the project code to the client, but only issue a non-exclusive worldwide licence to your utility code. Similarly, if you're a photographer, you still own the actual photograph, but may specify to a client how it's legally able to be reproduced. Plenty examples abound, depending on your line of work.
Mostly, I find written contracts are WAY clearer about who gets what and when, and shows the client is at least willing to "trust but verify". I wouldn't do business without them.
If you "work for hire" (see 17 U.S.C. § 101) an exception is made to the general rule that the person who actually creates a work is the legally recognized author of that work.
Requiring written contracts seems like a handout to lawyers. Alternatively, it seems like a clever way to enable folks with only a verbal contract to refuse to pay, since a verbal contract is now illegal.
Contracts are already enforceable at law, so how does this actually improve things?