I am not an expert on international law but I am 99% sure that the legal arguments in the text are partially wrong. Yes, some aspects of the UPC are not great but it should rather be opposed on public policy grounds, i.e. negative effects software patents and increased cost of litigation (and maybe that should've been done during the negotiations on the treaty).
It's a bit difficult to discern their argument as the text is filled with errors and, seemingly, some words were jumbled around.
1. Vienna Convention on the Law of Treaties (VCLT): The text of the treaty has already been agreed upon and states have signed (or even ratified) it. They are not negotiating the treaty anymore, so the UK (as former negotiating state) not agreeing to the treaty has no bearing on its entry into force or violates Art 24(1) VCLT. As the para. states: "A treaty enters into force in such manner and upon such date as it may provide or as the negotiating States may agree." This is just about how the date is set, i.e. either it is laid down in the text or the States agree - and that has already happened. Somewhat regularly, states will de-ratify treaties or exit negotiations on treaties and the treaty can and will still go into force.
2. Art 6 European Convention on Human Rights (ECHR) [1]: According to the case law on "tribunal established by law", the "law" part must ensure its impartiality and independence and, thus, primarily governs its composition (i.e. appointment of judges, assignment to cases (where latter cannot "solely depend on the discretion of the judicial authorities")) [2]. Note that, even in the quote cited in the article it is the object of the clause that "the judicial organisation … is regulated by law emanating from Parliament", i.e. it concerns their composition (also: "object" = aim of article which does not necessarily speak to the specific obligations). The composition and the independence and impartiality of the Court are provided for in Articles 15 to 19 (esp. Art 17: "Judicial independence and impartiality" [3]). Only beyond those rules (as well others), can the Administrative Committee establish rules of procedure (which may not contravene the treaty [4]), after consulting with the Commission on compliance with EU law (Article 41). Regarding fees, it should at least be noted that the treaty provides for the possibility of financial aid for SMEs in Art 36(3) [5].
[1]: Case-law guide https://www.echr.coe.int/Documents/Guide_Art_6_ENG.pdf#page=...
[2]: Quoting generously from the guide:
> 209. The phrase “established by law” covers not only the legal basis for the very existence of a “tribunal”, but also compliance by the tribunal with the particular rules that govern it (Sokurenko and Strygun v. Ukraine, §24). The lawfulness of a court or tribunal must by definition also encompass its composition (Buscarini v.San Marino (dec.)). It is the role of the courts to manage their proceedings with a view to ensuring the proper administration of justice. The assignment of a case to a particular judge or court falls within their margin of appreciation in such matters. However, to be compatible with Article 6 § 1, it must comply with the requirements of independence and impartiality (Pasquini v. San Marino, §§ 103 and 107). The judge assigned to a case must be independent of the executive,and the assignment cannot be solely dependent on the discretion of the judicial authorities (ibid., § 110).
…
> 212. In principle, a breach by a court of these domestic legal provisions gives rise to a violation of Article6 §1 (DMD Group, A.S., v. Slovakia, §61). The Court may therefore examine whether the domestic law has been complied with in this respect. However, having regard to the general principle that it is, in the first place, for the national courts themselves to interpret the provisions of domestic law, the Court finds that it may not question their interpretation unless there has been a flagrant violation of the legislation (ibid.; Biagioli v. San Marino(dec.), §75; Pasquini v. San Marino, §§ 104 and 109). A court which, without any explanation, oversteps the usual limits of its jurisdiction in deliberate breach of the law is not a “tribunal established by law” in the proceedings in question (Sokurenko and Strygun v. Ukraine, §§27-28).
> 213.The object of the term “established by law” in Article6 §1 is to ensure that the organisation of the judicial system does not depend on the discretion of the executive but is regulated by law emanating from Parliament (Biagioli v.San Marino(dec.), §74; Savino and Others v.Italy, §94).
> 214. Nor, in countries where the law is codified, can organisation of the judicial system be left to the discretion of the judicial authorities, although this does not mean that the courts do not have some latitude to interpret the relevant national legislation (ibid., and case-law references cited).
> 215.Furthermore, delegating powers in matters concerning the organisation of the judicial system is permissible provided that this possibility is enshrined in the domestic law of the State, including the relevant provisions of the Constitution (ibid.).
[3]: https://www.unified-patent-court.org/sites/default/files/upc...
[4]: https://www.unified-patent-court.org/sites/default/files/upc...
[5]: https://www.unified-patent-court.org/sites/default/files/upc...