The Democrats made sure to flood the US with illegals and then demanded States to issue them ID, so they could vote and made sure they knew who they should be voting for.
If the GOP was smart, they would do a amnesty for people that lived there illegally, but committed no crimes and had worked the majority of the 10+ years required to meet the amnesty. Those are the people that you want anyways. Then setup a good guestworker program where they can come in, work legally for a bit and then go home. That would push a lot of the Hispanic vote to the GOP.
That's a bold and frankly ridiculous claim unless you have some evidence of an institutional effort by the Democrats to practice mass illegal voting. IDs don't equal votes, and each state has it's own process for registration and verification. The
Heritage Foundation has a data set and map of illegal voting (which includes deceased voters, people submitting false absentee ballots, etc beyond just illegal immigrant voting attempts) and even including all the other types has a whopping...1620 cases (only 100 if you select the alien category specifically) going back to 1982, though that is only convictions. Almost universally the studies I've seen show it be effectively non-existent.
For some more details though, we have Republican states
Utah and
Georgia that conducted recent citizenship audits of their voter lists (outside of normal verification which varies by state but involves citizenship documents and/or cross referencing other databases at different levels of government to verify they are eligible to vote, and includes things like whether they're a felon, if they've died, if they've moved, etc.). Utah found 1 confirmed case of an illegal registered to vote and 486 people for whom they're missing paperwork out just over 2 million, and Georgia had 20 confirmed with another 156 they're missing records for, out of 8.2 million.
The republicans have controlled the white house, house of representatives and senate from 2003-2007, 2017-2019, and from last year to now yet seemed equally uninterested in doing anything about illegal immigration until recently, and then it looks largely performative. They went so far as to stop a bipartisan law based on what the Republicans were pushing for when Biden was still president. In terms of illegal immigrants in the USA, it hit a peak total estimate of about 14 million in 2023 (about 4.2% of the population). Which is outrageous, but in 2007 under a unified Republican Government it was 4.1% (12.2 million illegals, which dropped heavily after the financial crash). It seems the Republicans loved having illegals willing to work for cheap, and still do as there seems to be little appetite even now for cracking down on people employing them.
Bush wanted something like you mentioned, a mix of better security, guest worker programs, and paths to citizenship and has mentioned not getting it done as his biggest regret other than the Iraq war. I think we'd be in agreement that's the best way forward.
For context I have issues with mass illegal immigration (or legal mass migration in cases like our TFW and international student programs), the effects they have on suppressing wages, potential for criminals to abuse the systems, etc., but the notion that there's a mass illegal voting scheme is blatant conspiracy theory territory.