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Yes, they can backdate it.
This is all in insurance law. Your coverage legally ends on the date of the event that causes the insurance to no longer be in place, which is the date of resignation in this instance. The employer here is probably not the person being unreasonable, but rather, it is the insurance company that they have the deal with.
Because it is a small company, they have very little if any ability to negotiate a contract in their favor for their group coverage. They get what they get from their carrier. You've worked with previous companies that may have had better relationships with their insurance carrier and were able to define an employee coverage period to include the entire month of the final month of coverage, but that is not legally required in any state or federal law.
Insurance companies can legally define the coverage period to only include that time frame where the person actually is working there, so there is nothing legally incorrect about having the employment date and insurance coverage end in the middle of the month.
Yes, you are going to be forced to pay for half the month if you take COBRA. I'm actually shocked that this is the first time you've experienced this. In the present market, fewer insurance carriers are allowing the extended coverage period beyond the event date leading to the COBRA eligibility. This is actually the norm now.