Just like Hadron said, Rooting won't fix a physical hardware problem just provide a possible workaround. It certainly sounds like hardware issue to me. I'm assuming you've tried carefully
cleaning the buttons and ensuring they aren't stuck under the chassis edges? Ensured the plastic pieces are firmly applying to the small button actuators underneath?
(I've really only rooted one device, and after rooting I installed a stock Sense ROM based off a new OTA that would've made rooting later more difficult. Rooting was just my forced path to leave options open for later. I set it and forgot it.
Are the headphones/speakers powered, or have a built in amplifier? I asked because of the rare but plausible possibility of an electrical short that could have been sent into the device via the headset connection. The headset and volume button actuators are on the same daughterboard within the HTC One. Though an electrical short would have likely just fried the circuits and both buttons likely wouldn't work at all.
I know things are a bit different on the other side of the Atlantic there, but...
Did you purchase the device directly from HTC or through a local retailer/service carrier?
I know that HTC handles warranty claims for many of the localized retailers/carriers, especially outside of the local retailers purchase/return period, but I'd check with wherever you purchased the phone from about possible local repair/refurbish/exchange before succumbing and sending it back to HTC. Many places will just refer you direct to HTC, but if you stick up for yourself you may be able to gain some traction in a live setting rather than calling some call center and shipping the device off for weeks.