Her NOT paying for anything has never been brought up either. She only asked him for a place to stay, but she never asked him to support her financially.
Yes, the entire premise is about how weird the situation. It makes sense for the characters to be wary (Kyou herself even draws attention to this). The readers, on the other hand, should be able to see the tone, genre, and format & be able to tell where things are going. If this is your first 4-koma Slice of Life, they're 90% fluff with a focus on 1 or more characters just being cute and/or dumb
I'm saying she's an idiot that's just as attached to him as he is to her. It's operating on "soulmate/destiny" logic, which is a trope that's become more annoying the more I see it. Though, to be fair, people whose lives are completely absorbed by their jobs is apparently a common enough problem that it's a plot point in quiet a few series
1) This is the sob story that raises more red flags than the mundane lie she told. It sounds like she's trauma dumbing to get sympathy and pretending to refuse his offer to make it seem like that wasn't her goal. Also, introducing the daughter only after an offer is made absolutely puts pressure on him.
2) This completely changes the tone & format. There's no way you're getting all that into a 4-koma & unless the comedy is leaning on the darker side, there wouldn't really be room to make jokes with this setup.
3) The only thing really needed to fix the situation would be having her actually call him in advance. There would be much less pressure over the phone where he's not actually looking at them when he makes his choice
Neither prospects have been brought up, therefore the default interpretation should be "she didn't offer", because that's the one that requires fewer assumptions (which is to say "what we see is what we get"). I can easily come up with explanations, conversations and more that happened off-screen to make a story work, but at that point I might as well write my own series instead.
I am not saying that the author is trying to make this story sinister, I am saying they are doing a poor job of making this light-hearted. What's the point of bringing all this baggage into a narrative (they haven't seen each other for 17 years, the father is nowhere to be seen, we are lead to assume nobody else could help her, he is still hang up on her, etc.), if they don't want to tackle these? And how does the story look, if the adults in the room ignore all of it just to play house first? You don't need to tackle everything at once, but you need to make the setting believable. Right now everything is happening because he still has feelings for her, which is fine (at least now we know why he is just okay with everything), but doesn't exactly paint a pretty picture either.
Honestly, I don't think it's a great idea to have Kyou act this way. We are talking about a couple of adults who have dated before. There are plenty of justifications to be conflicted or not wanting to hook-up even if she has feeling for him (her having a daughter and wanting to put her first could cause all sorts of conflicts), but after you have already gotten inside his home, I'd like at least for them to act like they aren't teenagers about their stuff.
1) How is this a SOB story any more than the current setup? In the current version it's heavily implied she has nowhere to go, and has been having a hard time, otherwise the premise sinks entirely. A coincidental meeting just makes it less shameless on her part to show up with the intention of asking for help 17 years later after their last interaction.
2) This changes almost nothing. Kyou can still be a scatterbrain, he can still be in love, they can still have their vibes going, the daughter can still be a bit upset about the situation but happy her mother seems to be at ease around him. They would just start the relationship with some basic, healthy honesty, and neither would look or sound like they are taking advantage of the other. The story already has melancholic vibes, I don't see how my version impacts the setting in any significant way.
3) This would be a fine start, but I was working under the assumption that the author really didn't want that to happen, for whatever reason. That would also still make it weird he wouldn't ask more questions, while my scenario exposes a few more details for the readers and the protagonist.