"How do you include asexuality as a viable option in a romance game?"
I suppose it could be as easy as - after the MC explains to his/her romance partner that they’re not interested in sex - snipping out all the references to sexytimes.
The real problem I think would be to either have the romance partner blindly accept your asexuality (which seems unrealistic given that he or she is at least statistically likely to be sexual) or have the romance partner working to accept your asexuality (or, alternatively, admitting that the two of you are not a match and calling the romance off). It would take a good bit of extra writing to accomplish the latter. The writer must ask themselves if it’s worth the additional time and effort to accommodate this (and other) possible scenarios.
It’s a strange line to walk: the writer could end up alienating a great deal of the potential audience on one side and on the other side… well, let’s be honest, the writer will never finish the story if he or she is too inclusive of the possible scenarios. Even if they were able to finish such an all-inclusive story, the end result would be a mess of pandering tripe as bland as a breakfast of stale cornflakes in water.
"In games where you’re not explicitly said to have sex, then does it matter?"
No, I don’t think it does. If sex is not integral to the story-line and has been given no importance over the course of the story, then I believe it’s perfectly fine to let that part of the relationship develop in the reader’s mind. After all, we’re all biased, so while I will think “yep then we had loads of sex” others will think “yep and then we cuddled in a hammock on the beach” etc.
However, if sex is integral to the story-line and there is sexual tension between the one character and the MC, then leaving it unaddressed is just bad writing. Guenevere by @jeantown is a game that handles sexual tension and expectations really well. Simple and direct “I’m not into sex” or “I’m not into romance” works for this game because the story does not rely on either (even though it is strongly influenced by both). A story that does rely on a romantic or sexual factor, however, cannot get away with this because the story simply falls apart.