It's a matter of the insurance policy whether your coverage ends on the last day of your employment or the last day of the month in which your last day of employment took place. Either is legal and in my experience it's about 50/50 which way it goes.
It is entirely legal, and quite, quite common for them to cancel your coverage until they actually have your COBRA paperwork and your first payment right in their hands. What will happen is that now that your COBRA paperwork has been submitted, your coverage will soon be reinstated retroactively to the day of the cancellation.
The information you were given was incorrect. They have up to 44 days to send you the COBRA paperwork. Once you have received it, you have 60 days to elect COBRA and from the day you select it, whether it's day 1 or day 60, you have then 45 days to send them the first payment. All in all, that means it could be as long as 149 days (or close to five months) from the last day of your coverage till the day your first payment is received.
Suppose, for the sake of argument, that you decided at the last minute not to elect COBRA, even though you'd already sent in the election form.
If the employer were required to hold your insurance open until they'd received your check, they would now be liable for up to five months of premiums on your behalf that they were not going to get reimbursed for. It would be even more complicated if you'd submitted claims during that period. The law does not require them to take on that committment. Rather, the law permits that your coverage be cancelled until your check is received, as long as the coverage is reinstated retroactively and there is no gap in coverage.