No, it wouldn't need new textures and code.
Thanks to the PS4 Pro and Xbox One X most PS4/XBO-games already use dynamic resolutions. The textures are already made for 4K-resolutions, even if the current console hardware reduces the resolution most of the time.
A game with dynamic resolution in a range of 1440p to 2160p on PS4 Pro / Xbox One X could run in solid 2160p (4K) without changing any code. A game with unlocked FPS in a range of 40 - 60 fps on PS4 Pro / Xbox One X could run with solid 60 fps without changing any code.
Additionally Sony has patented texture replacements for older games: https://www.eurogamer.net/articles/2018-10-07-sony-patent-sparks-playstation-5-backwards-compatibility-rumours-again
Sometimes it surprises people don't see the obvious. I'm not saying better specs will improve the assets per se. But better hardware can make performance and resolution improvements (dynamic resolutions are common).
The X1X also made tweaks like 16x AF in content not optimized for X1X. I'm no expert on the subject but I believe they could make other visual tweaks that override default settings. They could even make emulated BC content as high as 8K.
I don't think we need a patent per se to replace textures. Numerous X1X games did it via patches, Borderlands Handsome Collection just got patches for that. Replacing textures is significant work so it seems that will only happen if the developers want to do it which is rare. That kind of work is what remasters are for.
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