Why is market cap / equity value useful?

As I understand, when a company goes public, they only issue a piece of the company to be listed for sale publicly (could be lower than 10% of the actual company).

So, for ex in the case of D&B (PLAY) who went public in 2012, they raised $94mm issuing 5.88mm shares. This is obviously well below what the actual company is worth, which is even less than a years revenue for them.

What usefulness is market in determining the total value of the company if the company only has 10% of itself listed publicly? That doesn't seem like a fair assessment of what the company is actually worth, considering that not all companies listed are 100% listed, or even majority listed on the stock exchange.

I just can't see how or why we are only valuing the publicly listed portion of the company and acting as if it's representative of the entire company?

Does market cap include all shares of the company, rather than just the public float? What usefulness is the public float - if a company lists 10% of itself publicly, and I purchase the entire lot of shares, do I now only own 10% of the company, or am I considered an owner of the company? Unsure how this would work in stealth takeovers where PE firms buy up all the public shares - do the shares need to comprise >50% of the company in that case when that happens?

Sorry for the uninformed question - I'm just a freshman and interested in this before classes start.

Thanks