Starting with the obvious, the legal profession is regulated in most jurisdictions and having a law degree, or something that ends up being roughly equivalent to one, is usually a requirement for qualification.
So, if we take "mastering law" to mean that you want to be able to practice it e.g. by representing someone in any courtroom then having a law degree (or equivalent) may be a necessary step.
If instead you simply mean that you want to be an expert, then having "intuitive, everyday reasoning" is not enough. You can't just reason your way out of every legal situation unless you have already learned the necessary background information and skills. That background information typically involves a solid foundation in the main points of law within "core" topics such as contract, tort, land, trusts, criminal, and constitutional and administrative law, as well as the general principles of how the legal system works.
Without that, how are you going to respond when a client, opponent, or judge, asks you what the legal position is or challenges an argument you've made? Are you going to pull out a book and spend a few hours looking up the answer before responding? An expert already knows the answer for some reasonably common subset of the law, or knows where (i.e. what statute or case) and how to look it up very quickly.
I would also challenge the notion that law is always "intuitive, everyday reasoning". Legislation can be complex and unwieldy, judicial decisions can seem counter-intuitive and strange (or rarely, downright incorrect but still good law), and outcomes can depend on subtle factors which may not be at all obvious to an untrained person. Indeed this is a common problem when discussing a legal point with a layperson - their idea of what seems to make sense is often the opposite of what the law says.
Aside from knowledge of the law itself, being succesful also relies on other skills. You need to have a decent grasp of the language to be able to come up with persuasive interpretations which favour your argument, or to make sense of other interpretations. If you are going to appear in court, you need to have good oral commnunication skills and be able to speak cogently. These are all things that would traditionally be learned and practice in law school.
Note that none of this means that you need to have a law degree per se, but it does mean that if you don't then you will need to acquire most of the same knowledge and skills in some other way.
Response to comment
You've asked for an example of where the law can be counter-intuitive.
In contract law, one of the elements for a binding contract is agreement. This consists of an offer made by one party which is accepted by another. Under the general rule, both the offer and the acceptance must be communicated to the other party to be valid. An offer can be retracted before it is accepted. However, in england-and-wales as well as other jurisdictions, an exception exists known as the postal rule (Adams v Lindsell (1818) 1 B & Ald 681). This states that if the offeree posts their acceptance to the offeror then it is valid as soon as it is correctly posted, not when it is received. This can result in some counter-intuitive situations. For example, consider the following chronology of events:
- A makes an offer to B.
- A sends a withdrawal of the offer to B by post.
- B sends an acceptance to A by post.
- B receives the withdrawal.
- A receives the acceptance.
If the situation were intuitive, we would have a single rule that would apply to any given action. Regardless of whether an action is effective when posted or received, there would be no agreement because the withdrawal came first either way. But because the postal rule applies only to acceptance, we have a valid agreement which was formed at step 3.
Other strange situations are possible. For example, B could post a rejection of A's offer on Monday and accept it by telephone on Tuesday before the letter arrives. A valid agreement is formed because the postal rule does not apply to rejections.
Potentially worse, Scottish precedent exists (Dunmore v Alexander (1830) 9 Shaw 190) to support the idea that there is an exception to the exception. B can post an acceptance to A (which under the postal rule is instantly binding), but then retract the acceptance via a faster method. If the retraction arrives first, then the acceptance is no longer valid. Logically this makes no sense - if acceptance took place instantly then at that moment in time there was a binding contract. It is also potentially open to abuse: e.g. you could accept by post a contract to purchase shares and then decide whether or not to retract the acceptance based on what the share price does before the letter arrives.
The postal rule is not something which would be intuitive to someone who hasn't already learned it. Yet, if you do know about the rule, it is easy to avoid it by specifying in the offer that acceptance must be received to be valid (Household Fire Insurance v Grant (1879), Holwell Securities v Hughes  1 WLR 155).