There's a lot to unpack here. The attempted translation you proffered doesn't come all that close, so I've broken it down as shown below:
People (who find) that 'labor/time & effort' bothersome
が [subject marker]
buy PET bottles of tea
explanatory/emphasizing の + continuous form of である（です）
Thinking this way,
rather than us actually buying 'tea'
the various necessities of labor/time & effort in drinking tea
we are buying a service that is saving us
explanatory/emphasizing の + です
People who find the effort of brewing their own tea bothersome buy PET bottles of tea, and thinking this way, rather than actually buying 'tea' we are buying a service that is saving ourselves from that necessary effort (that we find bothersome).
Your lack of inclusion as to what this 手間 is is supposedly 'brewing one's own tea'. If not, just substitute whatever that 手間 is.