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.

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. ;-)