How about this? The biological pressure is to have sex, and only indirectly to have children. The only direct pressure to have children (as such) is social (people asking about grandchildren, cultural values, tax rates, etc.). I'm not sure if I believe this, but I do want to distinguish the target of the biological drive versus the social pushing.
On the other side, why do you think there is more social pressure not to have children? As a (childless) married male in his 40s, I can say I've experienced a ton of social pressure to have children and zero pressure not to. I grant that this is anecdotal, but I'm not sure what you're basing your statement on - no evidence comes to mind.
The biological imperative is to procreate. I don't want to split hairs and separate the mechanism from the goal. It's true there is plenty of social pressure to have sex or to be more sexually appealing, short of achieving the goal to have offspring. It doesn't matter. Remove the social pressure and people will continue to have children. It's not like they're going to forget how. Biology will take over.
I think there's overwhelming evidence for the social pressure to NOT have children. China's one-child policy comes to mind, along with the trend for certain categories of women in developed countries who are delaying childbirth until long after they become fertile. The contraception industry alone is enough evidence, and I'd be willing to bet that most abortions are a result of social pressure than medical necessity.
Both types of social pressure exist, I'm just under the impression that more people are actively discouraged from having children than otherwise. Neither trump the biological imperative we feel, consciously or unconsciously, as sexual organisms to reproduce.
I understand that you don't want to split hairs, and I don't want to split them merely to be difficult (or pick nits). Nevertheless, I'm not entirely convinced that we can avoid the distinction between the desires for sex and for children. I remain unconvinced that the driving desire is for children (or procreation, as you put it now). My argument, in a nutshell, is that we can see that the desire for sex is primary and paramount by observing human behavior. The desire for sex precedes, outlasts, contravenes and cuts across the desire for children.
Beyond that, I don't think contraception is necessarily evidence for pressure to not have children, so much as evidence for a desire to control when and how many children to have. Certainly some people use contraception to avoid all children, but surely many more people use it in an effort not to have over large familes, no? As evidence for this (partial, because there are other factors like fear and permanence) consider that most people choose less reliable one-off contraception rather than a vasectomy or tubal ligation, even when both are readily available. China is one country, and in this regard somewhat exceptional.
You don't have to guess if you read my response. That said, I believe the same goes for women. The biological push is towards sex; children are a result (sometimes) of the sex. (You can argue about variance of intensity or frequency of desire among males and females, of course. I'm not sure what I think about all that, though my instinct is to say that a lot of the variance is cultural rather than biological. But that's neither here nor there, at the moment.)
Here's a way to play with this claim. Imagine you have a person (male or female) in the right age range. Would giving the person a child to care for by itself remove or diminish the person's desire for sex? I'm not so sure. (Note, I'm not talking about exhaustion due to taking care of a child - a clear side-effect, nor about the biological changes that occur after childbirth.)
Again, I'm thinking about this issue myself, but I don't think it's cut-and-dried male/female, as you imply.
Are you saying there's no biological impulse to have children? In my experience, both men and women have sex drive, but females often (males less often) have an additional drive to have children. Why would satisfying one impulse also satisfy the other? Even if I eat, I can still get bored.
As for the second sentence, I would not be surprised if the social pressure works in different ways in different places. I am in Norway where social pressure is high: the state sponsors it, would-be grandparents will of course nag, and it seems harder to be recognized as a complete person in the media and by peers if you do not have a family.