Only profits are taxed, not sales. So if you "reinvest everything" you are not going to pay taxes.
You can hire more people, buy other companies, rent more equipment, etc. The net result is that the company grows but has no profits. It doesn't work if it buys real-estate, for example, as that increases the company value and the owner then needs to pay for the increase of the value of the company. But then it's on the share value increase, and not in profits.
High taxes can incentive grow as spending money to grow it's less costly than to capitalize profits. That's why lowering taxes is seen as an opportunity to withdraw money and reduce investment.
There will always be some 'leakage' and taxes to be paid. Non-profits are generally exempt from all corporate taxes. If you don't care about putting profits in your pocket, they tend to be very tax efficient.
In the case of Ikea, a royalty on all sales (for use of the brand name) is paid to a low taxed for-profit entity. That's how he made his money. The larger the empire, the more royalties he gets, so the structure does make some sense. Not as perfect as running a pure for-profit entity, but probably one of the most tax efficient.
You don't start a business from the notion of funding the state.
I'm saying option a is better than option c) spend wastefully because you pay high taxes.
Your comment seemed to advocate for this third option.
I now understand your meaning to be "paying high taxes is not frugal" which I disagree with for other reasons. Specifically paying unnecessary taxes would not be frugal but a company can still be frugal and pay high tax rates. To put it another way I do not believe tax avoidance is a frugal necessity.
It's not necessary.. but probably a personally trait of frugal people. You reduce expenses.
You're absolutely right however that high taxes tend to incentivise reinvestment (so you can tax deduct). But too high taxes also deter entrepreneurship to some extent, because past a certain point a lot of people will feel they're working more for the state than for themselves.
I don't disagree with your second paragraph but I don't see how it relates to frugality.
If you legally find a way to reduce your liability to zero (i.e. like Kamprad did), then you're also being frugal? This obviously might not be entirely ethical, but it's extremely frugal. :)
He most likely hated wasting money, whether that was on excessive expenses or taxes -- it makes no difference. The money is gone either way.
I have a friend whose grandfather built quite a large retail business (in the billions). He said his grandfather was the same, including the tax avoidance. Grew up poor, made it, and hated wasting money.