I think I'd be okay without a backstory to their germophobia, mainly because I know several people (even a close family member) that are just wired that way to the extreme. No backstory, just purely irrational fear.
that's totally fair as it pertains to reality - part of it for me is, story-writing typically wants "neatness" for all its parts; real life is broad and chaotic and nonsensical, but narratives are built on structure, where things have reasons and justifications for being included (and when they aren't, we get terms like "plot holes").
So him not having a reason isn't "wrong", but it's an uncommon-enough characteristic to include on a person, much less a defining attribute of one of the two lead characters, that convention would suggest that some aspect of their arc would involve addressing it and working through it to achieve character growth.
That, and if it were just "a facet of his character", given that it's presented as a "flaw" this whole time on his part, that means he has to change, possibly unilaterally, to
match with the FL, who is being presented as "not in the wrong" in all of their engagements where she is pestering him in some manner and he's trying to fend her off.
Because she's not really respecting his space (she's doing better since the pool-cleaning episode, but even here, she's treating it as a joke, more than anything), and that creates an imbalance in the dynamic between their characters.
If there really is no underlying reason for his mysophobia, then it's basically saying "he has to change a severe behavioral condition in order to end up with the romantic lead", thus painting his compulsion in a purely negative light. If there
were a backstory justification for his mysophobia, then at least that gives substance to this flaw of his, and gives them both something concrete to address and try to resolve together, and would also justify the FL having to change her behavior toward him to "meet him partway", as it were.
In short - if he's got no further depth to him, that's fine, but it means that he's "wrong" and she's "right" and he has to change for her entirely. But if there's more beneath the surface in his case, then he's not "wrong", and they can change together in a more balanced fashion that sees a healthier relationship for them both.