Abortion wasn't the contentious issue that it is today back then. The entire pro-life movement was started by Republicans as a way to gain more political power, using Roe as a motivator. That's the reason why I doubt the Republicans will ever seriously undermine Roe.
This is absolutely fiction. The pro-life movement started as non- or bi-partisan and found its home in the Republican Party only really after those activists moved there.
The reasons anti-abortion activists became Republican are complex and multi-dimensional in the early stages, but it's absolutely WRONG to claim that Republicans started or even significantly supported the early anti-abortion activists. Those activists were largely Catholic and therefore also largely members of the Democratic Party at the time Roe was written. See, for example, Congressman Chris Smith, who was the Democratic chair of the NJ Right to Life Committee but ran for Congress as a Republican.
Abortion was also hugely divisive through the 1976 primary between Ford and Reagan. Betty Ford gave a famous interview in support of the right to an abortion, and her husband (though he claimed to personally oppose abortion during the 1976 campaign) continued to tell the Republicans to abandon focus on abortion until his death. The Republican establishment of 1973–76 still had a large and powerful contingent that was supportive of abortion, meaning your imagined astroturf campaign would have been impossible.
I would recommend you read
The Invisible Bridge: The Fall of Nixon and the Rise of Reagan by Rick Perlstein, which goes into a great level of detail on the transition of working-class social conservative movements to branches of the Republican Party, with more specific examples.