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I think I can now answer this, based on research I've done since posting the question.

The underlying issue is that it's mathematically impossible to make all keys exactly right with any one tuning. Equal temperament, which gives all keys the same (slightly wrong) pitch ratios for each interval, didn't come into wide use until the 19th century (some say later). Before that, a variety of temperaments were used. Under meantone temperament, some keys sounded very good and others sounded horrible. Even the playable keys didn't have the same ratios for all their intervals. Bach promoted "well" temperament, which made all keys playable, but not interchangeable. The regular use of equal temperament has worn down our ears to we don't notice that a "pure" fifth, which should have a ratio of 3:2, 

Composers in the Classical era, if they had a good ear for pitch, noticed that each key had a slightly different flavor because its intervals weren't quite the same. A major or minor third was wider in some keys than in others. This influenced their sense of how each key felt. 

Beethoven and later composers took leaps into remote keys, and by the late 19th century all keys were considered equally usable. Equal temperament became a necessity. A better scientific understanding of the relationship of pitch and frequency (especially Helmholtz's work) made it possible.

To sum up: Different keys really did sound different in the Baroque and Classical eras. Composers chose their keys accordingly.