If we're talking about the same thing, Cisco's "converged network" is a data vlan and voice vlan, so still isolated logical networks. Yes, it's on one cable, and in a world where a desk/office only has one ethernet port, that's a fair plan. However, it's a safer/wiser bet to have more than one loop per drop. That's hard to correct after the fact, but when designing an office, you'd be a fool not to.
You make it sound like copper costs thousands per year just to keep it conducting. That's absurd. 99% of the costs are in the initial installation -- and that's pretty low, actually, compared to the cost of switches, ip phones, etc. Once installed, it's almost zero maintenance. How often do *you* have to re-punch a patch pannel or replace a keystone? In my current office, I've not touched any of it since it was installed ~5 years ago; previous office is the same story. That's been the case going back 20 years -- since we stopped using 10Base-2 that is.
I have seen damaged jacks due to improper crimps, and idiots plugging rj11's into rj45's (etc.) To that I say, if you don't break it, you won't need to fix it.
poe-capable switches are dropping in price.
POE switches are f'ing expensive; much more expensive than a non-poe switch. And I suspect they always will be, simply because they can. They're a niche item. Those who really need them will pay the extortion. I refuse to do so, and instead use midspan power modules. (which cost about as much as an unmanaged, non-poe switch. esp. if you get them on eBay.)
As for troubleshooting... time-to-repair urgency depends on the environment. A phone in an air traffic control tower... that's a zero down time thing. A receptionist phone at a car dealership, to them it'd be Very Important(tm) but on the whole isn't critical; I'd put it at the top of my todo list below any other fires. It should be fixed by the end of the day. The receptionist phone at a doctor's office, however, is a bit more important -- people cannot book appointments, just showing up leads to a big mess... And in my office, if a software developers phone isn't working, well it's likely been broken for days before they even noticed, so a few more days won't matter. And what does it take to fix... in 99.9% of the cases with the VoIP phones (SNOM), they've lost their configuration. The remaining 0.1% is a "wiggle the cable" fix.
Many of the enterprises I've known don't like "new" and "current" technologies. They prefer "what we know" and "what works", and that generally means buying several year old devices to match what they already have. They can get very pissy when they cannot have it, too.
Continuing to support POTS is an interesting balancing act. Does a VG224 cost more than upgraded cabling, VoIP phones, and UC licenses? Probablly not. Do you really need a VoIP phone everywhere there's an analog phone? Again, probablly not. (fax machines, postal meters, conference phones, dorm rooms, etc.)