Always list the company name, not the 3rd party verifier. The company that I work for does not do its own verifications either. We direct employers to the 3rd party verifier. Many companies have outsourced this part of HR.
Second, most of my previous employers have closed up shop all together. They were small locally owned businesses and not reachable for verifications. Is this hurting my resume by having them on there?
If you cannot prove this, then yes. My sister once had to prove that she once worked for a bank that had gone belly-up. Both of us have our pay stubs going back ten years.
Your best shot at employment is through another small, locally owned business. The Fortune 500 types are just too picky today.