Thank you to everyone who responded.
I spent most of the day yesterday researching this. I think the problem is related to implementation of "early media" and SIP Code 183 messaging. We are not using SIP trunks; we are interfacing with the PSTN via a Level 3 (formerly Time-Warner Telecom) PRI. I think there is/are a configuration issue(s) related to this messaging somewhere either in the Level 3 service, the Patton gateway, the 3CX VoIP software, and/or the YeaLink phones.
This vendor lowballed the bid because, in part, they were not highly experienced in this type of work. They're a white-box computer and IT administrative services vendor who only recently got into phone systems. Ours is, by a few orders of magnitude, the largest phone installation they've ever done, and they had only done a literal handful before us. We trusted them because they've been very good on strictly computer stuff. Their management is failing to accept that this is different: They did not sell us a server, software, switches and phones, separately. They sold us a phone *system*, and they are responsible for all aspects of the system functioning properly.
Digital phone systems are also computers, and some of them, including the one we just replaced, also have separate hardware and software components. The fact that VoIP PBX software can run on a Windows computer does not, IMO, change the "business model" with regard to the sale of phone systems in any respect. A digital PBX vendor would immediately accept responsibility for failing to deliver a properly-configured system and fix it, for no charge.
I suspect that a more experienced provider would understand this problem and would not have had much difficulty in getting the configuration(s) done correctly. This vendor, due to inexperience, underbid the installation labor cost, and underestimated the difficulty of properly configuring our system, and is now trying to recoup its losses.
I sympathize with them, but I am not going to pay them to learn how to design and install phone systems properly.
Ken Dibble www.stic-cil.org