Although the visual impact might seem subtle on Guardhouse's first picture, it can get much worse on other backgrounds with more color intensity and micro details if that problem persists. "Main Street" is a better example where more prominent colors are too washed out or fading out, and important details are lost because of that.
I agree it's important to apply chroma denoising on FF9 (PSX) backgrounds and it's normal for color properties to change with resizing and filtering, especially on sources like this. But I believe that washed/fading out color problem in fine details can be avoided, giving your backgrounds better visual quality and fidelity. Your last picture doesn't have that problem so I think it'll be easier for you to identify what's causing it.
It is pretty hard to guess what are you talking about.
Flags on windows (or whatever that is) or flowers (if they are flowers), that sort of thing?
If so, this is known. From computer's perspective there is no difference between details and noise so just like any kind of image processing, this is a trade-off between removing what I consider bad and keeping what I consider to be good. It is practically impossible to tune processing in such a way that will fit all parts of every image.
Posted images may be a bit outdated, but they are made in a way that produces consistently-pleasant results for me across all images. Anything else requires manual tweaks (like limiting processing for certain parts ff the image) which I'm not doing much. For now this is a very low-priority stuff for me because I quite like the results as is. Huge amount of work on edges and animations concerns me much more.
But I believe that washed/fading out color problem in fine details can be avoided,
To put it simply, no it can not be avoided. An attempt to do so also exposes stuff that I do not want to see. On the main street image, except flowers and flags I do not want to see anything else on other parts of the image (not counting badly made sky that, I believe, was fixed since.)
Edit: you said that last picture of Guardhouse does not have a problem. Well, for me it does. The most obvious things like "lockers" look better indeed, but I don't like most of the other things. And this will affect some of the other images much more strongly. That said, most of the stuff that you call a "problem" is most likely made on purpose and/or known but not sure when or if it will be changed.