Whilst the points made by the OP are extremely valid, calling a business is a very 90s affair.
I would much prefer to use a medium such as Whatsapp to communicate with BA, underpinned by a fully-functioning website and App that is specced to handle transactions such as complicated itinerary ticketing, IRROPs for every eventuality (including rebooking on other carriers) and ability to perform upgrades, vouchers, jokers etc.
That said, I make do by using the email feature for contacting BA - communication is on my terms at the time I wish to communicate. It seems to work for now until more modern approaches are adopted by BA.