Its specs says it has Integrated quad channel SATA 3 Gb/s controller with RAID 0 or 1 capability therefore adding SSD won't make much improvement with SATA 1 controller. Its specs
https://support.hp.com/emea_middle_east-en/product/hp-xw6400-workstation/1844973/document/c00775249
But if you wish you still can use SSD .
>>> checks system compatiblity and worryingly, when I ran it, it said they do not have any SSD drives that are compatible.
Normal SSD's now use SATA 3 standard and they have form factor 2.5" that why Crucial says no SSD drives are compatible. As mentioned above, you still can use 2.5" SSD. There are many around, you don't need the best one because you won't gain much for your money and you need a caddy to convert to 3.5" format
https://www.ramcity.com.au/accessories/drive-brackets/2.5-inch-3.5-inch
Regards.