Any old sata SSD will work fantastically. The MX500 is a fine choice. The Samsung SSD offerings are good too. I prefer the 980 series myself.
Samsung seems like the gold standard, but the MX 500 seems more in line with what I want to pay on something this old. I was considering some like the Patriot Burst line or even some of the stuff on AliExpress, but some reviews I found scared me away from those.
You won't notice the higher speeds on the Samsung models. You might not even notice the speed difference between the Crucial vs whatever the cheapest option available at Micro Center, but this one is a bit tougher to say.
If you're ok with the Crucial MX500, it will be a huge upgrade.
Source: I upgraded a 3rd Gen laptop to the MX500 for occasional use while traveling. It's only recently that I'm considering buying a new one.
With an older machine, would it matter if I got a dram-less SSD? Would I notice a significant difference? I've seen a Teamgroup SSD (the T-force Vulcan Z) that's a bit cheaper, but it has no cache. It does appear to have a newer architecture though that appears to give it a longer life.
Especially because manufacturers threw shittiest HDDs in their laptops an SSD will do wonders
I have an MX100 from way back, and it's been fantastic in several older laptops. I love it, and Crucial makes some solid SSDs.
Make sure to download Crucial Storage Executive to get upgrades and to set over-provisioning.
ETA: You are correct that an SSD is one of the cheapest and easiest upgrades for an aging PC that will provide one of the most noticeable boosts in performance.