Xbox Series X is slightly more powerful, but PS5 has better exclusives.
However, most of those exclusives are for an older crowd.
With a PS5 they can play Spider-man games though, which might be a pretty big deal for two boys in that age range.
Xbox also has a subscription service with access to a lot of games. This is available for both an Xbox Console and PC.
PS has a similar service, but it's not as good.
Crossplay is so common that playing most things with friends shouldn't be a problem. Though I don't know how much online gaming a 7 and 9 year old might be doing anyway.
If it has to be a console, you can probably "get away" with an Xbox Series S for around $250. It's the less powerful little brother to the Xbox Series X.
Honestly, I'd just build a PC if I were in your shoes.
Not some "beast" of a machine, but something that will run most things at about the cost of a console and do a lot more, even if it's primarily for gaming.
It can easily be a media machine for whatever space it's in.
You don't even need a monitor if you have a TV and it's acting like a console, just hook it up like you would a console via HDMI or DP.
Just use a cheap wireless KB+M for desktop navigation. Then once you get Steam launched, hop over to a gamepad. It's not really any more complicated than that. Just be sure you lock down purchases just like you would with a console. Family View takes care of that.
It's for two kids so you can do a budget build, and it might be fun to get them involved with building it.
I wouldn't do more than a 4060, which is around $300. You can get a 7600 XT for around $250. The GPU will likely be the most expensive component.
You could go even cheaper given the age range. I'd go with AMD if you go any lower. They have a better cost to performance ratio. I'd get a 6000-7000 series card.
You don't need AM5, an AM4 mobo should do the job fine. AMD CPUs are cheaper and a Ryzen 5 is plenty for the sort of build. Something like a Ryzen 5 5600 is around $130 with a cooler included. (Don't forget to buy thermal paste.)
16 GB of ram is enough.
The rest just go for a good deal. You don't need much PSU for that, 450-500w. Around $40.
A mobo can be had for cheap, just get one with Bluetooth if you're using a console gamepad. Or you can just buy a cheap USB Bluetooth dongle.
It's not that hard to build, plenty of online resources with step by step instructions.
You will need to factor in the cost of Windows into the budget.
You'll need a flash drive to install Windows, and shouldn't need to "flash the bios" anymore in the vast majority of cases. Especially given the use case of this machine. That's mostly for people trying to squeeze out ever little bit of performance they can get out of a high end machine these days.
They can use a gamepad for PC gaming and don't have to learn to use KB+M, but it's also an option. Steam natively supports the Switch Pro if you've already got one of those.
Best gaming library available, cheaper games, digital through Steam/GoG so no disks, etc...
Don't sleep on Good Old Games, they sell more than just really old stuff, and there's lots of age appropriate stuff available there and on Steam.
Or, if it's in a communal space like the living room, you could buy the PC you want and say it's for the kids because of your "bet". If you're into that sort of thing.
Just putting it out there that you have an "excuse" here if you want one.
NPC: "Great hero will you do this task for me?"
Player dialogue options:
Game Devs: "Your choices matter and have an impact on the story."