In the Merriam Webster link, there is this entry under adverb:
— used to indicate that a number, amount, time, etc., is not exact or certain
The price of dinner was around [=(chiefly US) about] $50. [around=approximately]
It lasted around [=(chiefly US) about] a century. [around=approximately
The repair should cost around [=approximately, roughly] $200.
We should leave in around 10 minutes. [around=approximately]
We got home at around 8 o'clock. = (US) [around=approximately]
We got home around about 8 o'clock. [=it was approximately 8 o'clock when we got home]
To go to work at around some [time on a clock] is an adverbial phrase. It goes with (modifies they say in traditional grammar) the verb go.
Each of the examples given in that Merriam Webster's Learners Dictionary is also adverbial.
At + [time on a clock] is used as an adverbial phrase as it states when "go to work" occurs.
The preposition at is used with expressions of time. The use of at is idiomatic in expressions of time and associated with adverbial usage.
- It was about or around 8 o'clock =approximately modifies the time 8 o'clock.
Around or about is used in BrE and AmE and these examples from the BBC could be either:
We use around or about when we want to indicate approximate times:
When does his train get in? Around / About ten o' clock usually, but
you can never be sure these days!
What time should I come? ~ Come
about / around eight.
Approximately, expressed as around or about [a time on a clock or duration] is considered to be an adverb.
Expressing a time in English
idiomatic usage in English is: at a time
If a time is not definite, and is approximate, we can say: at around or about [some time].
at around eight o'clock means: at approximately eight o'clock. Approximately or around or about in these idioms are adverbs or adverbial. See below.
He arrived at around 8 o'clock.
around modifies the adjective 8 [8 hours of or on the clock).
It is around 8 o'clock.
- at around or around have the same meaning in time expressions.
You could say: that eight is an adjective modifying o'clock (eight hours of or on the clock), and around or about or approximately would be adverbs.
There are other interpretations possible, with at a [time] being viewed as a prepositional phrase. But there too: at around x hour(s) on the clock, it would still be an adverb modifying the hour: eight, two, three etc. o'clock.