Why is it not something you can just see at all times?
Why isn't it? Many banks and credit cards provide access to your credit score (and sometimes credit report) from one or more agencies as a perk. Some will provide you with periodic (weekly or monthly) snapshots, others allow you to query on-demand.
Same with the agencies themselves, all three have a free subscription option where you can sign up to monitor your records with that specific agency and check their value for your credit score.
The reason you cannot calculate it yourself is because there's no one formula. Each agency uses their own proprietary formula, and in fact some have multiple formulas that they use. These are secret and are not publicly available in order to prevent abuse and manipulation.