Agree that 'heard-only' can be just as valid as seeing birds - if not more so...a nightingale wouldn't seem like such a special bird if it sounded like a great tit, would it?
That said, I don't count 'heard-onlys' towards numbers on my life / year lists, but that is personal preference, as for me it is a list of birds I've seen. Equally, I don't count unidentifiable glimpses of birds confirmed on the basis of song or other people's ID.
Although not counting towards totals, I do record 'heard-onlys' on my lists, including songs I don't know that are pointed out by competent guides or other birders (I'm thinking of a pale-chinned blue flycatcher at Chitwan back in '89 here...). This must be common to many birders, as Scythebill provides this option. I also agree with jurek that sometimes its best to just enjoy the song / call, rather than causing unnecessary disturbance.
I think most birders understand perfectly well the difference between the personal experience of seeing and hearing birds and scientific / biological recording. There is absolutely nothing odd about wanting to see and enjoy a new bird, but if you're carrying out a breeding bird survey then counting the numbers singing is way more important. Equally, camera traps are valuable tools in biological recording, especially for species which tend to be shy and elusive, but in this case you have very clearly not experienced it yourself. To answer another hypothetical question, in the admittedly unlikely event of 'camera-trapping' a thylacine, I'd have no hesitation in sharing and claiming credit for my discovery. In those circumstances, I suspect the camera-trap evidence would also carry more weight than my personal account illustrated with a blurry photo through the trees.