I have a brother all-in-one, but pretty sure HP printers work the same. To receive or print, my phone app handles the interim processing and sends the fax/print job to the printer through Brother's internet services to my printer located inside my local network. The printer needs access to the WWW through the WIFI router however.
Who is running the interface between the web and the phone network? That’s what I don’t understand here. Such services, e.g. Google Voice, still require registration — they aren’t as free as the web itself.
Yes as you said, you do not need an all-in-one printer to send a FAX from your computer or cell phone or receive them for that matter. There are a number of services that provide this service, some of them for free, some with a paid subscription. For us, hooking the all-in-one printer to the “phone line” makes it work exactly the same as a normal FAX for sending and receiving along with being able to print and FAX documents from the computer.
The Keep It Simple Stupid principle works best for us in this case. If we want to send a FAX we generally print the document then FAX it. It is an extra step but we usually like having a hard copy in a physical file of the documents that we send to financial institutions, banks, and insurance companies anyway which are the only ones who we generally FAX to anyway.
I forgot, I have a “healthcare reimbursement plan” where I have to FAX multi-page documents to the administrator to get compensated. I typically scan the forms, all the receipts and create a multi-page PDF file and FAX that to them using the all-in-one printer through the WIFI connection. Otherwise I have to stand by the all-in-one printer and manually scan everything just before sending the FAX or tape the receipts to standard sized paper to get them to feed through the document feeder. It is a nuisance no matter how I do it.