I haven't seen "Anna and the King," so I can't comment on that. My other main problem with "The King and I," aside from using Thailand merely for "local color" as you mention, is that it takes away a lot of the agency from the king; rather than someone who actively decides to take control of the direction his country is heading, he's often portrayed in the movie as being rather backwards and ignorant. I haven't seen the movie in quite some time, so I'm sure (or at least I hope) that they do mention that he brings in Anna, but what I remember most are scenes in which she's basically educating him about pretty much everything. And the POV of the moviemakers seems to be on her side -- she is the light of reason in this backwards country.

On the historical side, it is extremely cool that Thailand was the only country in Asia not taken over by Europe and/or America. But I agree with coffeeandink above that Western exploitation in the area affects it a lot.