Resizing downward will make your pictures better! If you imagine every picture with a number of flaws, when you resize downward to 800 x 800 to reduce the scale of the flaws along with the picture content. The picture content remains but every picture flaw is reduced.
To answer your question, your issue is not a consequence of a cheap compact but more the resizing algorithm being employed in your post production.
Can you show some examples.
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