I think that historically in times and places where abortion has been highly restricted, abortions have taken place in spite of the restrictions. This is especially the case where travel to unrestricted jurisdictions is possible as, for example, it was from Ireland to the UK. But the historical evidence seems to suggest that abortion laws are not very effective. Laws that are designed to harass providers or to just make it difficult for women getting abortions (waiting periods, ultra-sound scans etc) probably have no effect at all.
Now, with medical (drug) abortions easily available and there being no prospect of effectively restricting the drugs the only result I can see from even the most restrictive laws on surgical abortion in the US is an increase in early medical abortions and travel to Canada.
I simply don’t understand why all this pro-life effort is directed towards the law and banning things that almost never happen (partial-birth abortions etc.) than towards advocating and supporting women to maintain their pregnancies and give birth into caring, supportive communities. I am not saying there is no such effort, just that the legal battle appears to me to utterly dominate despite its strategically gloomy outlook for pro-lifers.