First, you are correct, many plugins will fail if the header and footer hooks are removed. What your client is doing is going to be very problematic– dare I say “wrong”? That strips out a lot of functionality. If fact, there are limited dependable hooks on the front end and many are associated with the head and the foot.
Second, the snippet you have won’t work because that is a footer hook and the footer has been removed.
I don’t honestly see many good options for you. You can’t echo stylesheets after </head> or you get invalid markup (which usually still works) so most hacks to include stylesheets are going to be dicey. By hacks I mean “put do_action('wp_enqueue_scripts') and other required hook calls in your code after the point at which your code executes.
Otherwise, you will need to depend on inline styles and probably hack plugins on a case by case basis to force them to work in your very broken environment.