I agree with JS55.
If your employer hires the erector, your employer is responsible for their work. Should the subcontractor (the erector) drop the ball, the general contractor will look to your employer to complete the job, i.e., hire a different erector if necessary. Should your employer fail to hire another erector or if the first erector was "a nightmare", both the subcontractor (the erector) and your employer (the fabricator) could be discharged as not fulfilling the requirement of the contract.
The contractor (your employer) is responsible for quality control, both in the shop and in the field (because they hired the erector). The owner is responsible for quality assurance, i.e., to ensure your company did the required work, met the code requirements, performed the required inspections, etc.
Many states have adopted building codes that invoke requirements for special inspections. Structural steel framing usually requires special inspections that are performed by inspectors that are qualified (and certified in some cases) to perform special inspections as described by the building code. The special inspections can include shop inspections as well as inspections of the field work. The special inspections do not relieve the fabricator or the erector of their quality control responsibilities. The fabricator and the erector are responsible to follow their quality control manual (assuming they have such a document). Various certification programs such as the AWS Certified Fabricator and AISC's Certified Fabricator and Certified Erector's programs require the fabricator and/or the erector to have quality control programs in place. The third party special inspector may be performing QA functions to ensure the QC functions are performed by the contractor or they may have specific inspections tasks assigned by the owner that have to be performed.
In any event, it would be good practice (and it would receive recognition from the GC and owner) if your employer perform field inspections to ensure your subcontractor was "doing it right". The idea that your employer hired a subcontractor to perform the field work and to absolve your employer of responsibility for their work doesn't hold much water. Your employer is going to be held liable for any problems caused by their subcontractor.
Times are changing, what was accepted practice in the days gone by is no longer acceptable. When I started in this business, there were no CWIs and there was no field inspection. Then the building codes started requiring inspection of both shop and field work. Then AISC started their Certified Fabricator program that required the fabricator to have a quality control program with "qualified" shop inspectors. Now the Certified Erector program requires the erector to have a quality control program. So, overtime, the idea of quality control and inspection has matured and the idea that each entity is responsible for their work, i.e., the quality control of their work, has matured and become more formalized.
Best regards - Al