It’s not true that bastards never inherit – Alyn Velaryon did, Bran considers making Larence Snow the heir to Hornwood, etc. – but it’s extremely rare. So Catelyn’s not entirely wrong about the potential domestic politics in the long-term.
But she is taking the worst-case scenario. The Blackfyres are at one end of the spectrum, but the Longwaters are another possible outcome, and there are clearly others (as we can see from Westerosi heraldry).
I wouldn’t call Catelyn paranoid, because I don’t think her feeling toward Jon is primarily motivated by fear.