The melody of the song must fit the vocal range of the singer. If you don't know the word, look it up: tessitura. If you're accompanying a classical singer, there's no transposing. Singers choose material or productions choose singers based on suitability of range. If you're accompanying pre-rock popular vocal music, e.g. cabaret or vocal jazz, the singer chooses the key. Those songs get performed in many different keys. You'd better be ready to transpose to the key your singer wants, but you aren't trying to sound exactly like a particular recording. If you're in a (rock-era) cover band and the goal is to sound as much like the original recording, you would be wise to stick to the original key. That means you should choose your material to fit your vocalist, or choose your vocalist to fit your material. If you're in an original band or a band playing familiar songs with your own interpretation, you can transpose any way you like, but the performance is yours, not a copy of someone else's.
The OP is addressing the case of a cover band where the goal is to simulate the original sound, but the vocalist can't really do it in the original key. I tend to agree with the OP that it no longer sounds like the original. Something seems "off". It might make me feel a little edgy. You don't need perfect pitch to notice it. The bandleader has to make a compromise. His customers demand certain songs, and they demand that all the songs sound as much like the original as possible. There's really no other way to go in this case. Most of the audience will be happy with the compromise -- it's a wedding reception, they're drinking and dancing -- but those who are listening closely (and that probably includes the band!) might just be put a little on edge. Such is the life of a working musician.
