I have no love for Silencer Central, but I think what you are describing is consistent with the required process.
Capitol Armory is a very legit shop who also does direct-to-your-door suppressor sales. This is what their FAQ explains for those customers:
“If you would like it shipped directly to your residence, there are a few extra steps involved. We will have you complete a digital ATF Form 4473. Once you complete the form, sign it digitally, and submit it, we will notify the CLEO in your state that we are conducting a non-over-the-counter sale to you. We provide your contact information but do not include the details of your order. We are required to wait seven days from the date they receive the notice before we can ship your item. Due to the CLEO notification and seven-day waiting period, the shipping process may take 2-3 weeks from the date you complete the ATF Form 4473. Once the waiting period has expired, we will ship your item to the address on your ID/DL/LTC, and you will receive a tracking number for the shipment. We ship all items via UPS ground with adult signature required.”