As line faults and broadband problems often go hand in hand, what he is saying is that most line issues will get fixed in the course of fixing the broadband ones.
Your citing of broadband connections working when the phone doesn't is spurious. The broadband in such cases is severely downgraded. The cause of this scenario is one wire of the pair becoming disconnected somewhere. His cover level on the Business Infinity would come into play.
Also noted was MHC reply but it was not stated if the fault was with voice or broadband. Neither was the wait period for action stated but turning up in the early hours of Saturday morning is within BTs Prompt Care scheme promise.
I too have BT Infinity. A check with BT revealed that my cover is Critical Care on the exchange line and Prompt Care on the broadband service. My enquiry brought forward an offer of a 2 year fixed price contract for that telephone line of £14.20 per month. Interesting it was a fixed price contract, usually there are price rises even when in contract. I was reminded that the line rental will go up in August by a £1 [taking it to £19] - is this unwelcomed increase to pay for more engineering staff or for propping up BT Sport?
Line faults can indeed degrade the broadband service, so fixing the former has expectations of returning the broadband service back to its earlier operating status, but it doesn't always happens that way. Lots of moans about that have appeared on TBB. Also ,there have been instances quoted on TBB where the voice service has been lost [cut off?] and yet the broadband service continued to work perfectly for a considerable period of time. I therefore reject RobertoS claim that my observations were "spurious".
That aside, I am not sure that every ISP will offer an enhanced care service for either the exchange line and/or broadband service equivalent to BTs Critical and Prompt care schemes. As PlusNet are bedfellows with BT, they appear to offer some kind of similar cover to BT but exactly what that is I don't know. An enquiry to Zen revealed that only Standard Care was available. As broadband services for business becomes more "mission critical", then discovering that enhanced care of some sort is simply not available means greater care is needed in selecting an ISP.