"Is it the usual meaning as a noun (success / vicinity), or something else?"
You are thinking of another word 「[当]{あ}たり」 as you mention "success". "Vicinity", however, is strongly related to the word in question.
This one is 「[辺]{あた}り」. 「当たり」 and 「辺り」 are pronounced differently as well -- pitch accent-wise, that is.
This usage of 「~~~あたり」(It is more than O.K. to write it in kana.) may not be easy for the learner. It is a somewhat euphemistic way of giving an example and making a comment about it.
The structure is 「(an example) + あたり + (speaker's comment or opinion)」.
"Can it be replaced with something like のは without the sentence losing much of its original meaning?"
Good insight! That is exactly what I meant by "euphemistic".
If you said something like "~~が~~するのは~~だ", it could at times sound too straightforward and/or judgemental. By using 「あたり」, you can soften things a little without changing the meaning.
「こんな勝負を持ち出すあたり、~~~」 means:
"Her proposing a match like that, for instance, (makes me wonder ~~~~)"