No, capital costs are never as you list above.
You're not an accountant, are you? Items which qualify as "investment" are very specific...and quite limited.
The proceeds from stock offered for public sale by corporations can be used to pay off debts. The money raised through stock sales doesn't have to be paid back and often has fewer restrictions on company activity than other means of raising money. Or it can be used for any business expense.
The proceeds can also be used to pay the expense of the offering itself.
It also provides liquidity, potentially a way for venture capital to make a profit on its investment and as a means of compensating those who are granted options.
The idea that proceeds go strictly for "investment in plant, capital equipment, etc" (I have no idea what etc means to you in the context), is not true.
When one buys stock in a company one hopes it is an investment, but it is also a gamble.