It's a mix. Not every 401k administrator supports every feature, but the larger ones mostly do.
The benefits choices (including fees to the provider) are a negotiation, where the company has a certain amount of control.
If the company is in the negotiations discussion with other providers, that's the optimal time to suggest changes. If they literally just wrapped that up and announced the new provider to the company, that might be a much less flexible time. ("We just spent time and money on that issue; let's get back to our actual business now rather than spending more on 401K topic.")