Yes, in some cases, but this is not automatic, nor even close. The more serious the trial (ex, murder, child pornography), the more likely it serves the court’s interest to use the illegally obtained evidence. See https://doi.org/10.60082/2817-5069.3711 for a longitudinal study. Illegally obtained evidence is routinely used.

You used a conditional so I assume you also know how such a system can fail. It's not hard to figure out how that can be exploited, right? You can't rely on that conditional being executed perfectly every time, even without adversarial actors. But why ignore adversarial actors?

my understanding: within the context of that specific action; the evidence still exists. If there is less clarity about how and when it was collected though, there is far more opportunity to use broad evidence obtained in the periphery of a undisclosed warrant in other contexts.

Maybe. Courts aren’t magic machines that do the right thing.