Your argument proves his point.
Without birth control, procreation is driven by the extremely strong desire for sex. With birth control, that is no longer the principal reason.
The fact that the sex drive is SO strong is a clue about how important it is for “fitness”. However, with easy access to birth control, sex drive becomes far less important, and the drive to have children emerges as much more important (that is, to the extent that most births are planned – which would be the case if birth control were perfect.)
I think the OP has an excellent point, and haven’t seen a compelling argument against it. The drive to have children may well increase in the population, thanks to birth control.
Of course, that assumes that the drive to have children has a genetic component. No doubt there is a huge social component as well. But if there is a mutation that makes it easy for one to ignore the social pressures … I’ll let you fill in the blanks.
The arguments about birth control existing for ages is weak, because it hasn’t been terribly effective until The Pill. AIDS and the concomitant increase in use of condoms might be another watershed in the practice of birth control.

There is no evidence that behavior, at the level he describes, is genetic. There is, meanwhile, tons of evidence that it is societal.
Cite?
I suspect that there are and can be genetic factors that affect the tendency to want children. But, I’m a bit of a sociobiologist: I believe that evolutionary pressures can affect complex behaviors and proclivities (though not as specifically as many sociobiologists seem to think.)
There are definitely some mistakes in the OP, but if we look for the germ of truth, I think it’s clear.
So, to rephrase it: “Universal access to effective birth control is likely to produce a greater desire for offspring in the population.”
I haven’t done the math, but I suspect this would be true regardless of whether the population is growth-limited by resources (i.e., “Malthusian”).
BTW, there are other reasons Malthus has been wrong – so far. The main one is technology. His argument was that land cultivation was growing polynomially, whereas population was growing exponentially. Even if cultivation were growing polynomially, yield was increasing (or, started increasing just about that time) exponentially, thanks to technology. Technology has avoided every limit to growth, so far. We’ll have to see about that carbon footprint stuff. Fingers crossed!