I don't think anyone knows for certain if not paying upfront really does delay your process. Maybe this opinion was formed because some people did not pay it and they were delayed for some other reasons and they came to conclusion that this was the factor of the delay!
The payment form is electronic and I am sure if you enter your case file or UCI (how ever they call it) it will reflect your case so they will know and confirm that you paid!
Maybe I am totally wrong so please correct me!
messenger