The previous replies assumed that when one of your scripts "keeps a log of its output", this means it is always appending to a single log file every time it runs (so the log file gets longer with every run). If that's what you meant, then one idea would be:

 # check the size of the log before the main script starts
 # check the size again after the logging "sub" scripts are done
 # if the log has grown,
 #   open it, seek to the original size (start of new data)
 #   read everything that was appended on this run.
 #   filter out anything that you know does NOT describe an error
 #   mail the remainder to interested recipients.
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On the other hand, it occurred to me that the script that logs its output might just be creating a new log file on each run, in which case the same strategy would work, without having to check the log file size first or seek to the start of the current output -- just read the whole log file.

The important thing is to start by identifying and removing the stuff that people really don't need to see; if some irrelevant stuff gets through, that's less of a problem than failing to identify and keep all the possible relevant data. And it'll be easier to add to the set of irrelevant patterns to be deleted as people get tired of seeing them...