So is it generally better to have loads of private and public corps, rather than state and national corps?
Yes and no country run corps, don't pay the production tax to the country which is a MAJOR source of revenue for the majority of successful players. But because of this these corps can afford to spend this money not taxed on production to extreme salaries, which can set of a cycle of revenue to the country. Higher salaries are collected as income tax, also increasing corp production, increasing corporation revenue, allowing you to increase salaries a bit more (repeat until the corp breaks even).
And have the most of the economys backbone based on the private and public corps?
I've based my economy on public corps, I've found it to be successful, but it has drawbacks like all other corporate structure.
Anyway, should i hit the perform a public offering button?
You need to have a plan of whats is going to happen to the corp first. Are you turning the corp truly public (and what type of truly public corp), are you selling the corporation to Gamemaster enterprises. either of these two reasons will be beneficial to your country, but won't be able to bail out the corp so make sure you know what your doing. Any other outcome will not be beneficial in your case.