It seems to me that the left's argument is that there is no such thing as an illegal immigrant.
Ugh. Wrong, wrong, wrong.
First, there is a semantic debate. Some people -- of
all ideological persuasions, not just the left -- dislike the term "illegal immigrant." People are not "illegal," even when they violate the law when crossing a border. Thus, the preferred term is "undocumented immigrant" and/or "EWI" (entered without inspection).
Second, your "analysis" is flat-out wrong. There are plenty of immigrants who enter and stay in the US, without proper documentation, and never apply for asylum. Many of those individuals are entering the US legally, and overstaying their visa.
Third, requesting asylum doesn't ensure that the applicant gets to stay in the US indefinitely. The denial rate is around 65%, and if you are denied then you are deported pretty much immediately. Applicants basically stay in the US for years because there is a massive backlog for immigration courts. (If Trump really wanted to get rid of asylum seekers, he should be doubling, tripling, quadrupling the number of immigration judges/courts. Makes ya think....)
Thus, the options should be, at a minimum....
• Enter legally, stay legally (e.g. apply for asylum at a POE, leave when visa is up, or are granted residency)
• Enter legally, stay without documentation (overstay a visa)
• EWI, find a way to stay legally (successfully apply for asylum, marry US citizen, Dreamer, U Visa)
• EWI, stay unless/until deported (most undocumented immigrants in the US)
• Enter with or without inspection, remain in US legally until hearing held (asylum)