But the rod and staff don't require you to use them as a focus, you just need to hold them.
I think the rules are fuzzy on this one; after reading the items again you're probably right since it doesn't state anything about them being used as the spell focus. However, I still think it's a DMs call given the non-technical wording of 5E. Barring a Sage Advice column I don't think there is a single answer.
Having said that, an extra +1 or +2 isn't going to be game breaking considering the opportunity cost of using up attunement slots and that you're dealing with at least one very rare item.