By the way, you put >$300 as your budget which means greater than $300—did you by chance actually mean <$300 as in less than $300?
Do you have any storage preferences in terms of capacity, or even large-capacity mechanical hard drive (HDD) vs a smaller-capacity solid state drive (SSD)?
Presumably you want to use the laptop for things other than Dolphin? That'd certainly be a good reason for why a more dedicated gaming device isn't really a valid option for your supposed use-case.
Lastly, have you ever replaced or upgraded RAM?
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So I must admit, I'm not super-up on specific laptop models as, ever since the Framework laptop came out, it's sort of been a case of "all other laptops are dead to me", but they're definitely still too expensive for that budget despite the initial revision (Intel 11th gen) beginning to enter upper-budget range on the used market.
Honestly for the price you're looking at, it might just be best to try to find an older Thinkpad, or finding someone confirming if an exact model of laptop has good Linux compatibility (I found a reddit post for example regarding a specific Dell model).
The main issue is that, unless you can somehow find something with discrete graphics, you'll need to almost certainly rely on integrated graphics. AMD integrated graphics will be best, but AMD processors would be a poor choice for Dolphin unless it's Ryzen (or Zen-based Ahtlon CPUs), except those have the issue of not only being newer but also being sub-par choices regarding CPU performance relative to Intel until you get to like 4th or 5th gen Ryzen which would be even harder to get within your budget.
Therefore, in order to get good CPU performance for Dolphin without being stuck with newer, more expensive stuff, that can force you to go Intel anyway. And when it comes to Intel integrated graphics, especially of the older-ish variety, that can also force you to go with Linux which itself means a laptop that has good Linux compatibility—a Thinkpad being one of the go-to examples of such without going for more specialty brands like Framework or System76.
That being said, you'll probably be limited to just 1x internal resolution for older Intel graphics (or
maybe 2x internal resolution), and presumably not with even hybrid ubershaders or the like but you're welcome to experiment (if you're playing online, you may need to use the "skip drawing" option for shader compilation, ideally with "compile shaders before starting" also enabled).
One place you're welcome to get started on browsing if you care is Back Market (though you said you're not particularly savvy regarding hardware, so perhaps don't actually buy anything without asking first
) which I recently used for purchasing a refurbished phone and I've not really had any problems; your mention of a Chromebook makes me wonder if you're a student—they have $20 off any item if you're a student:
Also you can take a look at their item "condition" categories of Fair, Good, and Excellent—click the "Learn More" located to the far right of the "Condition" heading on this random much-too-expensive overkill laptop (because I couldn't directly link to the item condition definitions):