I found the problem, and it was not the SSD drive.
I spent some time watching the computer after it went to sleep, and I saw that it was actually frequently waking up and going back to sleep. I finally found the cause. The network adapter was configured as shown in the screenshot below. I fixed the problem by enabling the "Only allow a magic packet to wake the computer" option (or I could have disabled "Allow this device to wake the computer"). Only two things wake up the computer now: 1) me, 2) scheduled tasks like the nightly backup.