What makes strings sound "sweet"?

There is one thing no one mentioned. That is the type of violin and strings used in that recording. That will have a significant effect on what you here (obviously). The recording is part of it, yes, but I can tell you you'll run screaming if I place Dominant's run-of-the-mill strings on my violin versus Pirastro "Olive" Gold strings which are silver/nickel wound over gut and you'll melt in your chair.

The point of my response, while the OP is geared toward the recording, it still failed to recognize the instruments themselves. Some people really just play on crappy strings. It may have been the recording of the century too but it still won't sound sweet.

I think it is due to tone woods and the type of varnish they use and the strings and the bow and the player too.
How to get to Carnege Hall - Practice!

Of course woods and varnish have a role as with the overall size of the instrument as well. Horsehair bow vs. synthetic. Type of rosin used, ivory frog vs. plastic/bone frog, type of wood in the bow, etc..etc. The player is probably one of the more important keys. A good player can help a not-so-great instrument sound just a little better.

Citing all of this I think it is nearly impossible to obtain a perfect recording, unless you are Maxim Vengerov or Hogan. Then you are just perfect whether it be an iPhone recording or professional. ;-)

Good responses, but if we already have a well-recorded section (either live or a good VI like Vienna Symphonic or Hollywood Strings), how do we get a sweet Montavoni or Mancini sound? I *think* this is an EQ question and I am hoping for guidance.

Regarding the various harmonics in the series, I am impressed by the knowledge here, put I don’t know of a simple technique to alter harmonic content selectively.

A practical answer would be greatly appreciated.

*but