Taking a screenshot will save the raw 720p ps3 image. Doesn't reflect how it looks upscaled on your tv.
Let's clear something, some ps3 games look blurry because of the ps3 antialising.
Ps3 games that run at native 720p, witth (QAA) antialising will look slightly soft. Other than that, games that used other AA or no AA look very nice and clear. Games like uncharted, gta v, last of us...
Several years ago, i searched this topic thoroughly because i experience thr same thing as you just have. Usually what makes ps3 look so bad compared to Xbox 360 is the lack of a hardware upscaler, so you would have to depend onyour tvs upscaling. Most tvs have shitty upscaling, especially cheap brands like vizio, tcl oder low end models from other brands like lg and Samsung. Good upscaling will upscale without rendering the image blurry as if someone wiped Vaseline on the screen. A good upscaler on a tv, will upscale a 720p ps3 output and make it look similar to dlss performance or fsr, it won't look blurry, but different than native in a good way, i often admired this type of upscaled graphics and like it more than native resolution. It even fixes aliasing soemtimes. I can't explain it, you have to experience it with your own eyes. The second thing that makes the picture ugly, is digital processing, like sharpness and using dynamic image mode. Sharpness is usually high by default on most brands.
Here's how to get a good image from ps3.
Never play on a pc monitor. No matter how expensive it is, it was designed to run at native resolution. Upscaling there is basic and needed mostly when someone is installing windows or had issues with gpu drivers.
Avoid cheap tvs, the only exception is "Sony".
Don't use gimmic options like dynamic mode and options that enhance colors or black levels and learn how to calibrate your tv or at least get the best out of it.
Sharpness is different from tv to another, you have to find out what level is zero sharoness, it's usually 0. Cheap brands have minis sharpness and could get very confusing trying to find out which value is "no sharpness" for example when value gets lower than 20 or 10 etc... Image starts getting very blurry. You have to test this on native resolution. There are sharpness test patterns that can help.
Finding an old high end sony 1080p tv from ~2011~2014~ .will be the best way to experience ps3. Or get a high end/mid high end 4k tv, i prefer Sony, LG, Samsung. Believe me, it makes a big big difference. It's the same way for ps4 on 4k tv, most 4k tvs will make it look blurry, soft compared to native 1080p tv. But a very good 4k tv, will upscale it and make it look even better than it looked on a native 1080p tv. Ehat i am talking about cannot be photographed by camera or explained, you have to see it with your eyes. I have been through dozens of tvs, and experimented alot, sony tvs are best when it comes to displaying an image lower than native resolution. They're best at upscaling.