Birth control is not the only factor that has affected birth rates, even if it's the biggest. There are many factors. Any connection to COVID or climate change may not supported by any data you've seen, but does that entitle you to categorically rule it out as a factor? Where's this data that shows no relationship? I've seen a number of indications young people are increasingly hopeless about the future. It stands to reason that this could effect birth rates so I feel there's at least a small burden on you to show why the idea is wrong.
I mean it is possible somewhere in the rounding error or the rounding error of the effect contraceptives had there is climate change and covid effect. Massaging that out of the data will will be tricky to say the least, but given enough grant money I am sure people will rise to the occasion.