This is bad because?
The drawback to the amnesty giveaway is that you are importing people with no sense of civic duty and a vastly different culture.
A gradual decline in the birthrate to slightly below replacement level is not necessarily a bad thing. A sudden and precipitous drop to far below this point, as has happened in most European countries, is disastrous.
There are a bunch of problems. The most obvious will pop up 30 years from now, when a large number of retirees will be (not) supported by a much smaller and shrinking contingent of workers.
Combine this, in the case of Europe, with a large and growing group of unassimilated immigrants, and you are facing some major problems.
European countries, if viewed as organisms, are in the process of committing slow-motion suicide.
The US is in somewhat the same position, though to a lesser extent. Our white middle classes have a higher birth rate than Europeans, but what I think will save us is that we do a more effective job of assimilating immigrants. (Although our record here is not what it could or should be.)