That's a false analogy. A better one would be that when people come to our house, we welcome them in and see how long they're going to stay. If they don't agree to abide by our rules, we can ask them to leave, throw them out, or call the police. Your analogy is pretty close to the situation now! If you don't think there's essentially free access on our border now, you haven't been there.
I realise that is the current situation -- and our country is becoming a trash-heap because of it. Others here have said it better -- close down any business hiring illegal aliens; deport all illegal aliens as soon as apprehended; do not grant citizenship to babies born to non-US parents; shoot to stop permanently border-runners.
I don't invite everyone into my home, and I doubt you do either. That is why there are locks on the doors, and we turn away those we do not expect or have some type of relationship with -- even though that relationship may be superficially valid (a cop without a warrant).
Currently, there is (I think) a quota on how many people are legally allowed to immigrate each year from each country. And the right to legally immigrate isn't all that easy to obtain and takes a fair amount of time to get, which is why so many cross the border illegally.
We had a 'War on Poverty'. Then why are there more on welfare today than before the war? When you give things away, you eliminate the incentive to acquire it legally. You idea throws Citizenship out, and rewards lawlessness. You really believe that all those illegal aliens will tell the truth? That they can't hide once they get in-country? Please.
BTW -- I am an immigrant; Legal and Naturalized.