Can the publication conference choice affect one's admission to a new academic job?
I can't comment on whether the practice of explicitly pointing out that publishing in top conferences makes a stronger record is common, although I suspect it is taken for granted rather than spelled out.  But, from the perspective of a hiring committee, being published in a top conference generally indicates that


The research is excellent.
The work has been peer reviewed by other top researchers in the field.


Both of these indicators will give a hiring committee a better feel for a candidate's potential for future top quality work.

By publishing in a lesser conference or journal and without those indicators, a closer reading of the publication would be necessary to make a better judgement.

To comment on your question about whether the conference is more important than the work: for most people, I would assume (and hope) the work itself is more important.  But the reality is that hiring committees need to differentiate candidates, and judging by conference/journal quality is one way to accomplish that goal.  Additionally, trying to read all candidates' papers, and then trying to judge the worth of the research without necessarily being an expert in the sub-fields themselves, would probably be futile at worst, and frustrating at best.

Finally, strong letters in support of the candidate are probably almost as important as the research itself, especially if the letters point in particular to example publications that demonstrate outstanding work.