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I believe it is useful to look at it this way:
1.) Refugees and Asylees - These individuals have documentation of their entry and status. They can get work authorization. Can be a lengthy process. The bar can be relatively high. (If you want to watch a comedy that is based on a true story of a family seeking asylum, watch the Netflix series “Mo”)
Tragically, this can take a long time. Many with weak cases simply drop out of the process and remain in the US illegally.
2. Temporary Non-Immigrant Visas. Aliens enter to study (eg. F-1 visa), or to work temporarily (eg. H-2 visas to work seasonally in Agriculture, Hospitality, etc.). Some of these have a path to permanent residence, and others do not. Frequently this status expires, and aliens simply stay in the US illegally. (The 9/11 bombers were here legally on F-1 Visas. It’s a relatively low bar to get into the country, but converting to an H1-B upon completing education is non-trivial and typically requires for example an employer to provide sponsorship.)
3. Family members of individuals in one of the 2 categories above.
4. Aliens who sneak into the country

The people being flown from Texas/Florida are typically in category 1 above—not category 4. (Though category 4 can claim they are category 1 if apprehended.)