@barbdale: When I'm looking for a specific person, this type of search often _IS_ part of the very process you describe of "looking for enough background info". You start with whatever info you have (e.g., the example given of knowing a father's name from a death record), and keep looking for more. In the example given, I would also enter the known child's name in the family members list, and the spouse name if known, and locations they may have lived if known, etc. (This assumes I've already done all searches with the known child(ren) as the main search target.)
I also ALWAYS look for all related records on every possible match, because very often even when the match looks very good, it is someone else of the same name of similar age. There are literally hundreds, possibly thousands, of records here on Ancestry where I have added comments to the record of the form "this is Jack Doe who was born D1 and died D2, son of so-and-so. Do not confuse him with Jack Doe who was born D3 and died D4, son of blah-and-blah, who married Jill Brown. Proof is death cert which mentions parents and wife Jane seen here, informant son Harlan also seen here."
I do this when Ancestry's suggested hints show records for both Jack Doe's, and I encounter family trees where the wrong records been attached or wrong links made.
Also, often as in the death database search example, I am not even looking for a specific person, but for all records for anyone in a whole large family, which stretches the spread of ACTUAL event dates to wider than 20. It is a lot quicker to look for them all at once than to do 16 separate searches on each child in the family. I DO already enter the parents in the Family Members list, but with no date range specified, I still get matches that would be eliminated by using a proper date range.
I of course can quickly discard the irrelevant matches, and obviously that is what I already do, but it would be nice if they weren't cluttering up the matches list requiring more clicks.