Yes, it would work, and there are several reasons why neither political party in the USA will do it.
For the Democrats, it's obvious enough: Hispanics vote Democrat by a 60/40 margin, owing mainly to economic issues and not social morals (Hispanics are mostly Catholic and socially conservative, but pocketbook issues trump abstract social concerns for the poor). There are 11 million illegals in the US. Their children are citizens, and Democrat voters by a 20% margin. American population growth is robust, but driven heavily by Latino immigration and increase. Result: a relentless buildup of Democrat votes.
If there were ever an amnesty, the Democrats could expect to pick up 7 to 8 million voters at a pop, versus 3 or 4 million for the Republicans. And that would make the Democrats a permanent majority.
For the Republicans, it's a bit more difficult to see. There are three pieces driving them.
The first is economic on a business scale: big agriculture and big retail and construction, all three Republican ecomomic strongholds, rely on illegals to keep costs dirt cheap. The bottom line drives it.
The second is that suburban middle-class Republicans (and Democrats) work long hours and need/want domestic help for yardwork and day care. Latinos are generally more available, more malleable and cheaper than poor Americans. Further, there is an unspoken truth that suburban whites are much more willing to trust Latinos and Latinas into their homes and on their properties to do menial work and take care of their kids than they are to trust poor American blacks from the ghettos - the other available menial domestic labor pool. Given the high proportion of suburban and well-to-do people who have personal interaction with Latino/Latina domestics, there is considerable sympathy for them in that class of people - and these are the people who contribute money to political parties and are active in local politics. Take a step down the socio-economic ladder, and less well off working Americans who DON'T have domestic help are much more hard-minded about illegal immigration. But, of course, they're not the folks who man and operate the party apparatus.
Nowhere is the sense of closeness and even "family" made clearer than among the Bushes themselves. There are family members MARRIED TO Latinos. There is knowledge of these issues within the family. To the Bushes, the Latinos are not this foreign, invading race. They're related by blood and history and ties. American elites, Republican and Democrat, are more sympathetic in general to Latino immigration and Latinos in general than the folks further down the ladder. Politically, the suburbanites and elites will talk a good enough game to keep people from positively rebelling and walking out of the party, but they have no will or desire to really enforce any rules or do anything radical. Just give it time, and there will be so many Latinos that they CAN'T be excluded.
The third reason is a political calculation. Republican margins of victory are not huge. They are the majority party, but barely. Democrats get the Latino vote, but Republicans get the votes of third and fourth generation Latinos more and more. Since a very heavy Latino percentage is the future of the United States, actively undertaking policies that would alienate an electorate that Republicans NEED in order to hold onto their majority would not be wise.
So, the Republicans are riding a tiger.
They're not going to build a wall.
They will talk about it enough and take cosmetic public actions sufficient to keep the nativists on board for 2006 and then 2008, but they will never do anything effective. Anything truly effective will mean that the Latinos start voting for the Democrats 80/20 instead of 60/40, and that means a Democrat House, Senate and President Clinton.
There may be some such influence, but if so it is weak soup. Illegals may make the yard look better for cheaper, but that isn't driving any political influence or donations. The high income conservatives I know are universally anti-illegal, domestics or no. It is those for whom hiring illegals enhances income that drive Republicans, not those for whom illegals are merely a personal expense.