What you propose is not unreasonable, but it fails to acknowledge the present state of affairs. Under the Obama administration deportations were greatly stepped up and the flow of illegals across the border was brought to a trickle.
The VAST majority of citizens want better tracking of Visas and want illegal immigration kept at an unavoidable trickle where it is now. But the majority , not necessarily the vast majority, don't want those who over stayed their visas because of lax U.S. enforcement in the past, and encouragement by employers of illegals, and who made homes here, rounded up and deported. We want to look forward not backward. We want to be reasonable. We want to give those who have been here many years a reasonable path to legal status.
We must avoid buying into the demagogic rhetoric of our racist President, Donald Trump.
If those people you describe, over stayed visas, etc. are here illegally, then those are the exact folks I'm talking about.
Obama himself said "we are a nation of laws", those who are here illegally need to get legal. That requires effort from both parties, Government and the individuals. But if the individual isn't willing then sorry.
Looking forward, we should never have to face this situation again. And getting control of the boarder is the first step insuring that.