> What problems can you solve with NLP?

It's mostly understanding text and generating text. You can do named entity extraction, question answering, summarisation, dialogue bots, information extraction from semi-structured documents such as tables and invoices, spelling correction, typing auto-suggestions, document classification and clustering, topic discovery, part of speech tagging, syntactic trees, language modelling, image description and image question answering, entailment detection (if two affirmations support one another), coreference resolution, entity linking, intent detection and slot filling, build large knowledge bases (databases of triplets subject-relation-object), spam detection, toxic message detection, ranking search results in search engines and many many more.

I believe many folks are particularly attracted to NLP because the Turing test [1] is an NLP problem.

[1] https://en.m.wikipedia.org/wiki/Turing_test

Disagree. I think it's value is mostly unrelated to that.

All of the above, it's like asking what problems can you solve with math? HuggingFace's transformers are said to be a swiss army knife for NLP. I haven't worked with them yet, but the main fundamental utility seems to be generating fixed-length vector representations of words. Word2vec started this, but the vectors have gotten much better with stuff like BERT.

I thought transformers are mainly used for multi-word embeddings?!

There's a lot! Sentence detection, parts of speech (POS) detection to name a couple. These can be used to determine key concepts in documents that lack metadata. For example: you could cluster on common phrases to identify relationships in data.