No, it's not. The MSM is picking up on the exceptions, not the rule and turning those exceptions into the rule because they loathe and hate Trump.
As for our "immigrant heritage," we have NEVER as a nation accepted illegal or unregulated immigration any more than any other nation would. We also would not take criminals, the mentally ill, or those with serious or communicable diseases. In the Ellis Island years, any one of those three would get you into detention followed by deportation.
Uncontrolled, unregulated, endless, immigration has never been a part of US heritage.