Expecting enough of them to just volunteer their time doesn't appear to be a sustainable answer.
I spent a few years in IT as a Product Manager (or a similar role), and I viewed my primary role as protecting my team from the barrage of shit that I got, so that they could focus. This involved making sure I was politically the first point of contact and reducing back-channels (some are fine, but not ones that change functionality, involve significant work or are too distracting), placating the people requesting functionality or fixes by understanding how serious the dependency/issue was, triaging it and either placing it on the roadmap or saying no. We also had an engineering manager that could be the contact for specific bugs who could then triage and pass it on.
Some examples (that I have seen in reality):
Finance departments are cost centres, until you give them enough resource and they find you a more efficient tax structure. Cut finance departments start to struggle with things like credit control which affects your revenue.
Distribution Centres are usually seen as a cost centre, until you drop spend and it impacts COGS or customer lead times, or inventory in shops raises because of less frequent deliveries and you get out of stocks.
IT is a cost centre, but when funding is reduced change across the whole business slows and other areas are impacted (eg the customer web experience).
In reality the distinction of “some areas generate profit” and “some areas just cost” isn’t true in the end. All areas contribute to profit - some just do so indirectly.
I think the idea of Michael Porters “value chain” is better, where everything contributes to customer value (including indirect functions). The argument this makes is if you see some areas as just cost centres (e.g. fulfilment centres) then you can miss your ability to maximise customer value (e.g. offering faster delivery options).
Even sales people don’t usually generate profit on their own because without the other business areas they would be selling hot air.