A question for native speakers: has using the first verb gone out of fashion?
Yes, it has.
There is a sharp decline in its use after the 1940s, and the remaining uses seem to be those of the homonyms ("to be demolished", "to lay eggs" etc.)
After reviewing some online dictionaries I found the same definition, and browsing the internet returned hits like мы неслись по телефону
The phrase мы снеслись по телефону (sic) is an old-fashioned way to say "we had a phone conversation" (literally, "we got in contact over the phone" or "we connected over the phone").
I found one use of it online, in an article which is clearly written by an AI.
In what contexts is it appropriate to use?
I believe it has its limited use in officialese in the context of "foreign relations".
The Vienna Convention on Diplomatic Relations (1961) is Венская конвенция о дипломатических сношениях in Russian.
For a brief period of time in the early 1990s, the Soviet ministry of the foreign affairs was called Министерство внешних сношений.
Nowadays, both "diplomatic relations" and "foreign relations" are much more likely to be отношения.