Oh yeah, good idea. That should generally improve the results.
There's a bit of art to it for sure that can improve the results. You may also have to do some pronoun substitution in the summarized sentences (and then decide to do that before or after calculating the synthetic score) so they make more sense.
I find it's hard to tell how well all this will work until you just do it. Not every kind of text works equally well and the only proof that it's really working is "does the summary make sense or not?"
It's also possible that for long structured documents like laws or contracts, that you don't want to summarize the whole thing, but treat major sections like different documents and do intra-document summarization to maintain understandability.
Here's one that does something kinda like what I was writing about above.
http://textcompactor.com/
note I think the correct measure is not TF-IDF, but TF-ISF (Inverse sentence frequency) for single document summarization, but I might be wrong.