@chW33Nie There's nothing wrong with an all-in-one in theory. They're basically laptops with large attached screens: they use laptop hardware and are just as upgradable, which is to say not all that much. The problem with these AIOs is that none of them has a dedicated graphics card, meaning performance won't be anywhere near what it is with the laptops I linked. An Nvidia GTX 1650 is at least twice as fast in gaming as the very fastest available integrated graphics chips, with an RTX 3050 being about 30% faster than that, and these AIOs don't even have the fastest iGPUs currently out there.
The first desktop has the same problem: only an integrated graphics chip, and not a particularly good one at that considering the price and other options. The second PC has a GT 1030, which is about on par with the fastest iGPUs out there, although performance can vary significantly from one game to another and depending on which type of 1030 you get. (The faster model is faster than the best iGPUs in Sims 4 but still nowhere near a 3050 mobile's performance.) So for all these options, you're leaving performance on the table.
Please don't simply rely on reviews. People have wildly different expectations of a computer without a dedicated graphics card than a computer that has one, and the products are for two different consumer bases that don't necessarily have a lot of overlap. Even the same person would likely evaluate the two types with starkly different criteria, for example simply expecting a standard home or office computer to be stable and not too slow in basic tasks while wanting a gaming laptop to run demanding games with good performance. As an example, the fact that a laptop overheats and/or downclocks when running games would be annoying, but even in those cases, it would still be running those games much better than these AIOs, and cooling trays are cheap.
If you do want a desktop, yes, you can use any monitor you want, although it's best to stick with one that supports one of the connection types the computer supports as well, for example HDMI or DisplayPort. This is generally not a problem with any recent model, and you could get an adapter for an older one in theory, but in practice you might run into issues with the wrong adapter or a very old type of connector.
As for gaming desktops, I looked at half a dozen sites and only found one new option with a graphics card at least as fast as a 3050 mobile. That would be this one, which is significantly faster and on sale:
https://www.hp.com/us-en/shop/pdp/hp-envy-desktop-te01-2265tbundle-pc
I can look further if you want, but I wouldn't be surprised if other sites had similarly meager offerings in your price range.