This misunderstands how employer-provided insurance works for most people. Large employers sign up with a company like Cigna to provide a network and administrative process. But the actual healthcare is covered by the employer. So really, Cigna or BCBS don't really give a rip if you're taking a bunch of money out of the pool.
USA: That’s the case if your company is “self-insured”. Some are, some aren’t. I imagine there are financial requirements to self-insure but I’ve never looked.
Most likely, if you’re in a medium to large company (not sure the cutoff, probably somewhere around 500-1000 employees). Smaller companies will generally actually need the insurance company to be the payer as well since otherwise one or two huge payouts could bankrupt them.