Maybe it is just anecdotal, but working with hundreds of hispanic students over the course of six years, I saw a vast amount of immigrants learning English and taking pride in doing so.
Students, as in children? If they're forced to be in an educational institution where they learn English, then yes, of course they're going to learn English. Although they may not have been born in America, they would be more akin to second generation immigrants that are more often than not bilingual according to studies.
If you're referring to adults that you're teaching, then it's still no different. I'm not saying that absolutely no immigrants take the time to learn English. I'm saying that, by and large, first generation immigrants do not ever learn English well, and some never learn it at all. Again, the studies show this. Linguistic assimilation usually occurs over generations.
...speak English well.
I don't think the expectation is for them to become professorial in their speech. The majority of Americans don't actually speak English well.
As long as they can communicate basic needs and receive basic commands, I consider that a success.
Again, my entire point was that immigrants back then assimilated linguistically just as they do now: over generations. We can argue about how well immigrants learn the language over time, but it's clear that a child of a first generation immigrant is going to grow up with English and generally know the language way better than their parent.
The claim by GH was that people used to assimilate into America, and now they don't. Linguistically, this isn't the case. Immigrants back then continued to use their native language and did not take the time to perfect their English, just as they do now. The only time that we see real improvement with immigrants' English speaking skills is as their descendants are born and learn English. The first generation immigrants still speak broken English, and sometimes don't know anything other than the basics like you mentioned. Linguistic assimilation still by and large occurs over generations; the "good ole days" weren't a time when immigrants bowed down to American superiority and studied their asses off just to "assimilate."