It's hard to offer a reasonable approach to handle this without fully knowing your girlfriend's perspective. So let me give you a broader overview.
First, yes, there is a component of cultural misunderstanding. Ukranians, like Russians, are not culturally western but have a Eurasian culture (a mix of both Europe and Asia). So as an indian, I can culturally relate to her view of the typical household she describes and can offer you a perspective on it.
Broadly, in such a marital relationship the division of labour is:
- The husband is expected to be the earning member who protects and provide everything for the family.
- The wife is expected to have the responsibilities of fully managing the household and providing nurturing care for the family members.
If both partners are willing to this, it's actually a pretty reliable and good setup for raising a family - the earning member can fully focus on his career, without needing to worry about daily household chores or kids, which is the wife's responsibilities. Everyone in the family gets home-cooked meals, thanks to the wife. Kids grow up with one parent always there to watch their progress, in the early developmental years, and not feeling alone or neglected. Raising kids can become a community activity when the kids are younger as many home-maker wives plan social activities together - kids learn valuable social skills. As the kids grow up, and become more independent, the women gets more and more free time for her personal hobbies and / or social activities. Both husband and wife make some sacrifices for each other and for the kids.
Overall, you provide a stable environment for your kids to grow up in. (Assuming, ofcourse, that both parents are psychologically healthy individuals :).
While today the role a a wife as a home-maker is demeaned, anyone who has managed a household knows that it isn't an easy or cushy job at all. And that's where the kicker comes - if the man doesn't have a good profession, and doesn't make enough money, the wife's responsibilities becomes more stressful - if a husband can't provide her with a washing machine or a dishwasher or a microwave (or even a part-time maid, if you are a middle-class, in my country) that can make her job easier, she is going to be burdened with more physical chores. If you can't provide her with a car, she has to waste time travelling on public transport. All this means that the wife will have less time for her own needs and her family members.
That is why, it must be clear, why there is a cultural view that a man who doesn't provide a reasonably comfortable life is seen as someone neglecting his wife and family, by not earning or sacrificing enough for them. (From that perspective, you can see that the rosy picture of a man always relaxing and having nothing to do because his wife is a home maker, ain't really that rosy at all. :) Similarly, a women who doesn't uphold her own end of the bargain - to manage the household and the kids responsibly is looked down in such a society.
First decide if you like the idea of such a setup, where the divisions of responsibility are clear cut between the husband and the wife - the husband is the sole earner, and the wife manages the household.
If you don't like it, you have your answer on how to handle this. You need to tell her clearly that isn't the kind of setup you envision for your marital life. That you see your marriage as a partnership where both members work and financially contribute for the family, and both members share household chores.
If she doesn't agree to that, you have to make a decision whether to compromise or let go.
Otherwise, from her offer to pay you once, I assume she has a job. In that case, if she really expects you to take care of all the household finances, without any contribution from her, then it is not at all culturally insensitive or unreasonable to ask her to assume the full responsibility of household chores (and if necessary to work only part-time or quit her job).
It's good that you are realising the difference in financial values both of you have so early on - different financial values is one of the major reasons why marriages fail.
To sum up, decide the kind of married life you see for yourself. Decide if your current girlfriend can fulfil it (without suppressing her own desires). If yes, ask her what kind of marital life she seeks. If both of you can come to an common agreement and an amicable and acceptable understanding, good for you all. Otherwise end the relationship immediately.
(Note also that attacking your manhood in an outburst, when you were trying to share an important concern with her, indicates that she has an anger problem. This can create a lot of communication problem in the problem and will not be healthy for your relationship in the future.)