I am told that this is what most countries faced with such a situation do. This doesn’t rush their ultimate naturalization any faster.
The last time I know of was 1986 - when the US, for the 1st time in its history granted amnesty, disguised as "Immigration Reform & Control." Washington sold Americans that this pathway to citizenship (for 2.5 millions) with tough enforcement will end the invasion of illegals.
Quess what! It didn't end the invasion but invited its expansion to now 12-20 millions. And you and the politicians you support are trying to again sell the same lie.
You say, registration "doesn’t rush their ultimate naturalization any faster."
Who says "ultimate" naturalization should be guaranteed to illegals? And how would that stop the continuing invasion of tens of millions of would be illegals who have not yet headed to our shores?
Before 1986, America, under Hoover, FDR, Truman & Ike, conducted massive repatriation programs which deported hundreds of thousand and more importantly forces millions of other to flee back home.
Here's just the numbers for the Truman & Ike:
Truman(1945-52) 127,000 deported & 3,200,000 forced to flee to avoid deportation
1950s 150,000 deported & 3,900,000 forced to flee to avoid deportation
Source What America had traditionally done when "faced with such a situation" is mass repatriations that yield about 25 illegals fleeing home on their own for every 1 we deported.