The first control decision is whether the outside party provides any part of the trust service through subcontracting, outsourcing, or another third-party arrangement. If it does, EN 319 401 clause 7.14.3 keeps overall responsibility with the TSP for conformance with the supply chain policy, information security policy, and trust service policy requirements.
This turns the subcontractor file into assurance evidence, not only procurement paperwork. The TSP should be able to name the outsourced service part, the affected trust service policy requirements, the supplier-owned activities, the TSP-owned controls, and the evidence that proves those controls were communicated and monitored.