RAW - Probably not.
The staff creates a swarm of :
...harmless flying insects...
Thus, there are 2 characteristics of the insects created: they are both harmless and flying.
The second Giant Insect ability is limited to:
...up to ten centipedes, three spiders, five wasps, or one scorpion...
The only one of these that is flying are the wasps. It would then come down to an interpretation of the word "harmless" and as a DM I would probably rule that wasps would widely considered to be harmful insects (especially when compared to the harmless gnat-like creatures I would envision as being created on a plain reading and imagining of insect swarm. A swarm of wasps are also included in the MM with a CR of 1/2, so I don't think they would be "harmless").
RAI - Probably not.
Here it would again come down to the word "harmless". In the grand scheme of harms in a magical DnD world, wasps could be characterized as harmless, especially since a level 1 character would presumably be able to kill wasps without taking much damage (if any).
As such, I could see a DM being open to an argument along these lines, though ultimately I would probably still rule against it. The language describing the Staff of Swarming Insects likely would have been clearer if this was indeed the intention of the game designers (eg "you may turn up to 10 of these insects into a giant version of any of the following: 10 centipedes..."). It is also important to remember that a swarm of wasps is dangerous to lower level characters (especially level 1 characters). So while I would rule it could not be used in this way, it would ultimately be up to the DM.