Does "show" for "put in an appearance; arrive" sound any more or less informal/slack than "show up" in modern day English?
Solution 1:
I (UK) only use show, meaning 'attend', as a verb in negative and interrogative constructions (e.g. He didn't show. Did he show?). I don't use 'show' as a verb in affirmations (e.g. He showed for his appointment). Instead I often say 'show up'. In fact, I usually use 'show up', meaning 'attend', for all purposes (negative, interrogative and affirmative) (e.g. Did he show up? He showed up for his appointment. He never showed up for his appointment).
I use 'no-show' (noun) to describe people who do not come to an event (e.g. "There were a lot of no-shows. Is there anything else on tonight?").
I never describe people attending an event as 'shows' (or 'show ups').
I use 'show up' (verb) informally. If I want to sound formal I say 'attend', or 'go'.
Solution 2:
In British English it would be unusual to use show by itself (in this context) in the middle of a sentence, but informally it could be used at the end of a sentence.
He failed to show for his appointment. Wouldn't be used, sounds American.
He failed to show. May be used with "show" at the end, but sounds quite formal.
He didn't show up for his appointment. Most likely wording for informal conversation.
He was a no-show. Even less formal.
Note: I just spotted this is tagged with american-english. I can't comment on how formal/informal it sounds in American English, so this may be more useful to others finding this question than it is to the OP.