I had written a long response to you and then my system went down before I got to the submit button. Here's trying again.
Traditionally, when an employee leaves the company for whatever reason, the employer cancels their insurance coverage effective the last day of work or the last day of the month (whichever is stated by the insurance contract - in your case the last day of work is the last day of the month so it's immaterial) and, if coverage is elected, coverage is reinstated retroactively so that there is no gap in coverage. Assuming that your company had more than 20 employees and was dealing with Federal COBRA instead of the state continuation policy, they did a couple of things wrong, even though cancelling your coverage effective the last day of work was not one of them. They should have given you 60 days to elect coverage, and 45 days to send in your payment. They are not required to reinstate coverage until they have your check in their hands.
I have a vague idea where the July 31 date may have come from, but will have to do more research on it. Towards that end, though, did you have access to other coverage (which perhaps you did not take because you thought this coverage was still in force) and if so, did they know it? Perhaps through a new employer?
It's possible, assuming the company is still in existance (you said something about reopening) that you might be able to hold them liable for the bill, given that you have the July date in writing. The insurance carrier is correct that they cannot make payment without the company's say so. Ask the US DOL (they cover COBRA issues) about the company's liability.