Do clients become subject to the EU/UK GDPR solely by virtue of using Remote’s services and hiring in Europe?

No, they don't. See example 15 in EDPB Guidelines on the territorial scope of the GDPR:

A private company based in Monaco processes personal data of its employees for the purposes of salary payment. A large number of the company’s employees are French and Italian residents. In this case, while the processing carried out by the company relates to data subjects in France and Italy, it does not take place in the context of an offer of goods or services. Indeed human resources management, including salary payment by a third-country company cannot be considered as an offer of service within the meaning of Art 3(2)a. The processing at stake does not relate to the offer of goods or services to data subjects in the Union (nor to the monitoring of behaviour) and, as a consequence, is not subject to the provisions of the GDPR, as per Article 3.