Raid1 improves your recovery time for a drive failure.
RAID1 doesn't improve recovery time, it entirely eliminates the need to recover from a drive failure at all because your system never actually goes down. It saves your butt if downtime costs you money, and makes it so you don't have to restore OS/data from a backup. Nobody is making claims that it protects you from all the other possible causes of data loss.
Should you?
Absolutely not.
While it's not an ideal situation performance- or value-wise, that doesn't mean NOBODY would be better off using mixed RAID rather than nothing at all. And depending on the system capabilities, an M.2 SATA drive could be used resulting in both drives performing equally, or the cheapest, slowest older-model NVMe drive available could be used so that the performance difference isn't so big and not as much money being spent on unused performance.
Personally, no, I would much rather get a system that is capable of holding matched drives, even if they were both SATA, but if that meant paying much more (including considering the costs for the drives) or having to get a form factor that didn't work for them (as OP seems to want the small size), then the value might not be there for some people.
Im looking at older generations 5th-8th because of price point (I don't want to be over $200 USD). That being said I believe most of these machines in the above parameters will run Intel chips. I don't really know what it means to 'use Windows or Linux software RAID' VS 'If you want to use the system chipset RAID (Intel or AMD)'
Keep in mind the requirements for Windows 11 if you're using Windows and don't want to use Windows 10 after support goes away. Microsoft has already changed the code once where CPUs that could run older builds can't run the most recent builds at all even with workarounds because those older CPUs don't support the necessary instructions, and they could change it again later. They already are making half-hearted changes regularly to prevent installing it on unsupported hardware using workarounds, and could eventually go so far as to hard-code blocks.
Windows includes the capability to configure RAID arrays via Disk Management, using built-in RAID software, where the OS and the CPU do all the work of managing the array. (Linux and others include similar functionality.) You see all the drives in Disk Management and select the ones to include in an array and configure it. The OS copies the data to both drives in RAID1, splits the data up to stripe it across drives in RAID0, or does the striping and parity calculations for RAID5. That requires some amount of CPU time to do the work (a lot of CPU time for RAID5), and with older and slower processors with fewer cores, it could cause a noticeable performance impact. In ages past, you'd only use software RAID if you were running a multi-processor server and absolutely couldn't afford a hardware RAID card (they were very expensive) and were okay with slower performance. But an 8th-gen Intel with 4 to 6 cores like you're considering won't have an issue with RAID1. Even a 5th-gen with 4 real cores should be fine with just 2 drives, although it might start to lag if you were running an array with a lot of drives.
Chipset RAID like Intel's or AMD's, or cheap add-in controllers from Marvell and others, is called HostRAID, often called FakeRAID because the manufacturers call it hardware RAID but it's not really. The chipset doesn't perform the actual processing of the RAID data. Instead it uses software drivers in the OS to perform the same functions that OS software RAID does. The drivers and the controller hide the individual physical drives from the OS, and only present a single drive in device manager and disk management. Performance is largely similar to software RAID, since the CPU is doing the calculations via the drivers. The biggest difference is that with HostRAID the array configuration is stored in the chipset, so you could switch to a different OS without needing to rebuild the RAID (of course if you're wiping the OS, rebuilding the RAID isn't such a big deal unless you have data stored on a separate partition from the OS).
The nice thing about Windows software RAID is that you can plug the drives into ANY Windows machine and the RAID will be recognized immediately, with no need to install any manufacturer drivers. If your motherboard fails, you can just replace it with a completely different model and boot it up, rather than needing to make sure you have the same chipset, or you can move them to a completely different computer build as an upgrade. (Though this could run into the same issues as moving a single drive to a new computer/motherboard, in having cruft leftover like drivers from the old machine, but Windows is pretty good about this these days.) With RAID1, you can even take just one of the drives to plug into another machine and it will be read without issue, although Disk Management will show there is a broken RAID array with a missing drive, again with no need to worry about having the same chipset. (If you put that drive back into the original machine, you would have to tell Disk Management to re-sync the drives.)