There are several different legal systems in the UK; there are both national differences (as between Scots law and the law of England and Wales) and applicability differences (as between private law resolving disputes between private persons, administrative law resolving disputes between a person and a statutory/governmental body and criminal law wherein the state prosecutes alleged wrongdoers). All of these fit broadly under the term "common law", so that term needs to be disambiguated.

EULAs and TOSes are firmly in private law, and we can take England and Wales as the national setting.

Even here, "judges discover the law through principals and precedent" is inaccurate. First and foremost there is overriding statute. Where Parliament has intervened in matters of private law, Parliament wins; the parties may choose to show that Parliament's intervention does not apply for some reason (e.g. it conflicts with a subsequent intervention by Parliament, or it does not apply strictly in the matter before the court). Judges may act sua sponte, but mostly in private law leave such matters up to the parties to draw to the court's attention. Secondly, there's the plain wording of the contract. Finally there's recourse to covering case law established by higher courts and binding on the court of first instance (e.g. the county court or the High Court).

However, Parliament has caused the Civil Rules of Procedure for England and Wales to bind the county courts, and CPR rule 1 is the "overriding objective" which directs judges to be just taking into account the totality of circumstances and the behaviour of the parties, among other things. The UK Human Rights Act 1998 also requires courts to take into account the rights it brought into force, and this applies to all courts. These two features oblige judges to look past statute (or more strictly speaking, to do a reading-down as necessary) and specifics of a contract when assessing liability.

The private law system in England and Wales is (mostly) adversarial with the judges (mostly) paying attention to issues brought up by the parties' advocates. There are specific obligations on the court to act sua sponte as noted, and a court is free to ask questions or consider points not brought up by the parties, and it is also free not to look too deeply into matters of its own volition. This can lead to "judge roulette" to some degree, but the court-appearing legal community in England and Wales is not that large (and it's even smaller in Scotland or Northern Ireland) and good advocates and even good solicitors have some idea of what to expect from a particular judge in terms of case management.

However, I don't think many would agree that judges should "discover the law thorugh principals and precedent". Certainly almost no senior English judges woudl agree with that idea; indeed, the majority is much more likely to say that the parties should draw to their attention every salient aspect of the dispute so as to reduce the court's workload (in principle to do sufficient work that few disputes really need a hearing or a conclusion other than an out-of-court settlement between the parties).

They "discover the law" mostly by having it brought to the attention by the parties. Except in constructive litigation, the adversarial principle supposedly guarantees that one party cannot wholly misrepresent the law to the judge (unfortunately this is often not the case, especially where one party has much deeper pockets than the other, and even less the case when filings are not even dealt with because the cost of litigation exhausts one party even where that party has a good case that the non-exhausted party is misrepresenting the law).

The law stems from several sources. Depending on the area of practice of private law, statute and secondary legislation may have codified many aspects such that no other source of law is required in most cases, or (as in landlord-tenant law) statute law may be highly scattered across many Acts of Parliament, and additionally almost always engages in references to decisions by the Court of Appeals taken to resolve disputes where Parliament has not decided to provide a statutory basis for the resolution. (That's mostly because MPs are terrified of legislating in the area of property law since it is a daunting task to consolidate hundreds of years of various sources of law into one Act; not-so-jokingly the Great Repeal Bill proposed as part of the Brexit process will probably be less involving.)

Scalia's argument is overly idealized and focuses on the legal aspect of the system of justice, to the detriment of the justice side. A system of justice should lead to a finding of liability on wrongdoers, but should hold non-wrongdoers harmless from liability. (Unfortunately there are several aspects of the system of justice in England where that falls down, but at least there aren't many professionals in the justice system who think it should be even less just, finding non-wrongdoers unjustly liable simply because that is what the law says to do.)

You seem to somewhat contradict yourself here. On one had you speak of overriding parliamentary authority, on the other of Parliament steering away from "complex and various sources of law".

But your focus on advocates bring the law to the judges attention seems to support the parent comment, judges "discover" the law. Certainly statute overrides all, but the point of common law is that the statute is always insufficient. It is not enough to deal with the facts of any given case. I don't know much about the UK legal system, but in the US rulings on statute become codified as "precedent". Important and relevant decisions are published, circulated, cataloged, studied and effectively become the law. Any case that is litigated starts with a series of briefs on what the parties feel is relevant case law. It also might include briefs filed by interested parties, studies of the legislative process to determine intent and so on. That is all very much in the realm of "discovery".