Would you speculate that at one time, non-reproductive species existed, or would you say that reproduction is truly the order of the day? And if so, you didn't quite answer my question.
I don't think it is possible for a non-reproductive species to exist. The occasional individual, perhaps a family...but a species? I can't even begin to speculate how that would ever happen. Of course there are plenty of species that at one time didn't get to reproduce that went extinct - but species are a classification of life and the entire purpose of creating organisms is to serve as vehicles that propagate genes so a non-reproducing species seems to me to be the height of impossibility.
Things either survive, or don't. Survival seems to be the goal, but perhaps it is only the accidental result?
As I said - survival is the goal of genes. Organisms only need survive long enough to propagate those genes.
What I am asking is, even if you could say that survival 'pressures' are evolved as part of a species and continue to be part of their make-up, how would we explain the 'desire' of the first living thing to reproduce? Is there some gene which designates this drive?
Life is replication. The first thing that lived replicated itself. If it didn't replicate itself there are no other characteristics it would have that could have itself recognized as living. It would have just been organic chemistry. It was replication along with inheritance of traits that defined the beginning of life.
There was no 'desire' to reproduce, it is just that the first thing that reproduced and passed on that property to its offspring would have been the first thing we would be tempted to call life. Naturally its offspring would also have the reproduction property and so they would reproduce. If any of the resulting offspring lost the technique of replication that line would go extinct.
I'm sure in in modern complex life there are many genes that have been perfected to increase the chances of reproduction. One could almost say that is the entire use of all genes. Some genes aid in reproduction by increasing the chances of the organism in surviving predator attack, some aid in reproduction by increasing the chances of the organism surviving to find sufficient food and some aid in attracting mates.
As for the 'sex drive', in more complex animals, I'm sure there are genes that control the brain which is the thing that makes the decisions about how to prioritize time, and the parts that say 'we should mate now' have undoubtedly got a team of genes behind them.