Because illegal immigration has been a bigger "problem" for many years (and appears to be on the decline a bit), I always assumed that there really wasn't a desire on either side to address it.
I figured that perhaps the Dems didn't care much because that's the kind of shit they do. But the Republicans' failure to seem to give two shits about it always flummoxed me. All I could guess was that various business interests rely on illegal immigrant labor, and therefore the Republicans would not oppose. Or, closely related, the economy is fragile and dependent on that source of cheap labor. Thus, no Republican president wanted to start a recession on their watch just to address the illegal immigrant issue.
Either way, I've had the sense that for at least 30 years, neither party had much interest in ending illegal immigration.
In fact, this a large reason why Trump rose to prominence, IMO. The Republican base was probably as flummoxed as me as to why their leadership didn't seem to make illegal immigration a priority. Along comes Trump, and it is his no. 1 issue.
Now, I distinctly remember debates around the edges about what to do with the ones that are already here, but no one seemed fired up about stopping it from happening in the future.