I applaud your instincts.
With clauses like this are rare in speech; they are organizing devices employed primarily in the written language. The role of with is not very specifically defined: all it signals is that the clause it introduces is syntactically subordinate to its 'head' clause.
In my opinion it is employed most happily to signal an expansion of the immediately previous content. For instance:
At work we usually go out to lunch together, with each of the group selecting the day's restaurant in rotation. —The with clause describes the mechanics of the daily sortie named in the first clause.
Eateries compete strenuously for customers, with some handing out flyers in the streets and others offering lunchtime specials.
But your example 1 is to my mind an illegitimate use. Bare at work usually means "in the company where I work"; if a more general application is intended it means "in all or most companies". In either case, the second clause shifts the focus from what happens at work to what happens in the commercial environment beyond the office. Consequently, with does not clarify the relationship between the two propositions, it merely tacks on a new topic.
So I think your rewrite, 2., superior to the original. But there are many more devices for linking these two sentences; which you use depends really on the underlying relationship between them.
In downtown [CITY], where office workers usually go out to lunch, eateries compete strenuously for lunchtime customers by handing out flyers in the streets and having special offers at lunch time only. —Here I subordinate the first clause because it explains the superordinate second clause.
At my office we go out to lunch, and eateries compete strenuously for our business by handing out flyers in the streets and having special offers at lunch time only. —Here I coordinate the two clauses because you and your colleagues are equally involved in both.