This is a good idea and how far you have to go to find untagged systems was recently covered by this thread:
https://forums.frontier.co.uk/threads/some-data-how-far-to-go-to-find-a-new-system.521297/
As a result of that thread, I have been doing exploration close to the Bubble recently (within ~300 LY of Sol). I have, rarely, encountered untagged bodies. Mostly these are icy or rocky ice worlds and the odd gas giant 50+KLs from the jump-in point around a non-scoopable star. Completely untagged systems are much rarer but I have found a couple.
If your intent is to find and completely map these systems, I suspect you are more likely to discover The Dark Wheel's secret base than Raxxla. According to the Codex, The Dark Wheel's base is hidden in orbit of the 8th moon of a gas giant. (I'm not sure if the image in the Codex is actually it but there's a separate thread discussing that). I have certainly found untagged 8th moons. Also, a potential host moon only needs to be unmapped, not untagged, to be hiding the base.
Raxxla is a very unusual case, as it might not appear on the FSS. There may be as many opinions as players as to what it is. I think your search would only be able to rule out that it is not discoverable via FSS in the search region. It might, for example, be millions of Ls, or even several LY, into deep space and only appears if you are close to it. Given the time frame for its discovery, I suspect it is within 100LY of Sol but it needs precise directions to find, with The Dark Wheel's alleged toast possibly providing a constellation-themed set of directions but lacking the start system and distances that need to be travelled.
Good luck!