It's been primarily the ACA subsidies the whole time.
The bit about immigrants is that lawful immigrants could recieve Medicaid/medicare but the BBB
removes some of those immigrants.
ETA: In my opinion, it's a bad time for the Dems to be fighting over the ACA subsidies. From a strategic perspective, they should have reopen the government after a week or so and kept on message about how the GOP refused to keep them.
Come ~June/July when prices jump up, then they can force the issue. People would be more likely to support it once their health insurance actually increases.