Pre-built is not the same as OEM, so make sure the terminology being used is right. If you bought a machine that was shipped and built by ASUS or Dell or Lenovo, then it's OEM and will have all their own branded parts, as much as possible other than the CPU of course. If it was just normal retail parts that were assembled by a PC shop and sold as a standard assembly that they made many of, then it's just pre-built, but there won't be a "model number" that does you any good. You'd still have to be looking at the manufacturers of the individual parts for BIOS/firmware, drivers, etc. Machines like PowerSpec and many custom-build units you can get online are still just pre-built machines, not "OEM" because they don't actually make the components. They're just buying retail parts in bulk and adding margin to them, and charging you to put them together. Some of them do have custom case designs but that's about it.
If you're only changing drives because your current one is getting slower than it used to be, the problem may be that the drive is too full, or that it's having problems. If you get a new drive that is at least twice the capacity and clone the old one to it, you'll probably restore performance to where you were originally. SSDs lose significant performance, like 75% transfer speed reduction, if they don't maintain at least 10% free space, preferably 20%. Run Crystal DiskInfo and see what the drive health is reported as. Also make sure that TRIM is enabled in Windows. (Open the defragment application and make sure that all the drives are being "optimized" on schedule.)
Looking through the supported CPU list, I would say a Ryzen 5 5600G or 5600GT (newer, slightly faster version) is a better option if you're going to upgrade the CPU. It's a Zen3 processor (4600G is only Zen2), so you're getting the highest generation model your board can support. Zen3 is also still fully supported and AMD is even still making new CPUs with it. The 5000 chips also seem to be a bit cheaper since there is still wide availability. (The 4000 series with graphics was also OEM-only, so you'd also have to by a cooler. The one you have MIGHT work okay but might not.)
Do you have RAM that will be able to support the new processor? The Athlon 3000G uses DDR4-2667 while the 4000 and 5000 series use DDR4-3200. If your RAM isn't rated for 3200 you might be able to overclock it a little but getting it up to 3200 will be too much. It will still work for the new CPU but it will be at a lower speed than is desirable, taking away a lot of the performance that you paid for. Buying a new CPU and new RAM starts to reduce the value of trying to maintain this old system, but it depends on how long you expect to need to stretch it out.
Do you think you'd be able to do a full upgrade in like a year most, which would cost at least $600 for low-end stuff? If so, then I'd say save the money and stick with what you have for now, and only upgrade the SSD to a larger SATA drive. That's assuming the performance has been "good enough" up to this point. How much are you hurting due to the low performance of the CPU and graphics? Are you actually wanting to do more but can't because of that?
If you're pretty sure you're going to be unable to spend more on it for several more years, then $200 to $225 for a CPU and RAM (32GB) plus the cost of an NVMe drive would carry you through until then (assuming none of the aging components fail). You'll end up with what feels like a new system compared to the original build, even if it's relatively outdated and slow compared to the newest setups and will be very old by the time you have to replace it.
For the record, I'm on a 5600X myself right now, which was an upgrade from the 3600XT I originally had, on a 5 year old X570 board. It's still plenty fast, with an RX 6600 GPU and good NVMe drive, though I have to fight the constant urge to replace it just because there's newer, faster stuff available so I might see a second or three faster loads of applications or games. I think this system could work just fine for me for a few more years in terms of my actual needs.