“〜を” with adverbs, such as “入学を条件に”

This is something I sometimes encounter, every time I encounter it always confuses me for a second but it’s easy to figure out what it means every time, such as in this case “on the condition that he enter our school”. I’ve seen many others such as “病気を理由に” which again obviously means “by reason of illness” and it seems to in general be fairly interchangeable with “〜の”.

Is this a general thing that can always occur with adverbs or only with specific adverbs? And is there some kind of nuance difference? Could I also use “私と食べるのを条件に” opposed to “私と食べる条件に”. For instance I could find a citation “レビューを書くのを条件に”, how does this differ from “レビューを書く条件に”? Surely I can’t use “食べたのを後に” instead of “食べた後に” right? Surely “初めて会うのを時に” is also not acceptable for “初めて会う時”?