In doing a bit of searching, I found that there's been court cases brought to the EU high courts related to CSI. There seems to have been a ruling that states that, in some circumstances, having access to the NHS is equivalent to having CSI. Lifting liberally and editing for space, from a lawyer's page:
"The preliminary ruling of the Court of Justice of the European Union in C-247/20 VI v Commissioners for Her Majesty’s Revenue & Customs concerns the interpretation of EU law in relation to the requirement of comprehensive sickness insurance cover (“CSI”) in the host Member State."
"The UK’s position has so far been that access to the NHS is not sufficient to meet the requirement to have CSI and that economically inactive Union citizens (students or self-sufficient persons) and their family members needed private comprehensive health insurance (alternatively, a European Health Insurance Card for persons who did not intend to stay in the UK permanently, or to be covered by reciprocal arrangements)."
and
"Having established the interpretation of the Directive and Article 21 TFEU regarding in what circumstances and to whom the requirement to have CSI applies, the Court turned its attention to how the requirement can be satisfied. It confirmed that the requirement to have CSI is satisfied when the child has CSI that covers the parent and vice versa. The documents before the Court showed that VI and her son were affiliated with the NHS; the UK’s public sickness insurance system offered free of charge, in the Court’s words."
"
The Court confirmed that it is open to the host Member State to make affiliation to its public insurance system of an economically inactive Union citizen subject to conditions, to ensure that the citizen does not become an unreasonable burden on the public finances of the host State, provided these conditions comply with the principle of proportionality. The maintenance of comprehensive private sickness insurance enabling the reimbursement to the State of health expenses incurred, or contributory payments to the public sickness insurance system were given by way of examples.
Notwithstanding this, the Court unambiguously confirmed that once a Union citizen is affiliated to such a public sickness insurance system in the host Member State, he or she has comprehensive sickness insurance within the meaning of Article 7(1)(b)."and
"In practice this ruling could mean that, even if a person did not previously have a right to reside in the UK under EU law solely on the basis of not meeting the requirement to have CSI, once they are registered with or otherwise access the NHS, the requirement to have CSI is satisfied and the person acquires a right of residence. Whilst there are charges imposed for some types of NHS treatment to people that are not lawfully or ordinarily resident in the UK, there were no charges for individuals that had a right of residence in the UK under EU law (when that was applicable). Such an obligation to pay for NHS treatment would, thus, have been extinguished immediately by the very fact of having accessed the NHS, provided that that conferred a right of residence by satisfying the CSI requirement."
and
"The question of whether the requirement of the first limb of Article 7(1)(b) is also met, namely that the person has sufficient resources for themselves and their family members not to become a burden on the social assistance system of the host Member State, is distinct and has to be considered, in accordance with the principle of proportionality. The Court confirmed that, in circumstances such as those of VI and her family, access to the NHS free of charge, in and of itself, does not amount to an unreasonable burden on the UK’s public finances. Nonetheless, this will be fact specific and in different circumstances, a different conclusion could be reached."
So I think that some serious legal consultation would be in order before I bought CSI or went without it. If I was an EU citizen, which the OP is not. (I don't think.)
REF: 01 Apr 2022
https://immigrationbarrister.co.uk/comprehensive-sickness-insurance-csi-cover-and-the-nhs/