Methinks your understanding of how sacrificial anodes work is correct.
Many/most electric tanks are glass lined, and as long as the liner is intact the anode isn't doing much even if it's slowly giving up it's mass to the water. It's protecting the exposed areas where the electrodes & plumbing of-necessity pass through. Once it's gone there is still time (several months to a few years, or even several years, depending on the volume of water used and the water chemistry) before rust-through is likely. The fact that you got the old one out without a lot of trouble and it wasn't rusted-on is a good sign, but not a guarantee that it doesn't have an imminent rust-through risk.
But the fact that they also burned it up by dry-powering it in combination with the 100% used up anode puts it in the "please don't bother" column- it's not worth fixing.
In my area it's pretty easy to find "retired-working" electric water heaters from other peoples' rehab/remodel projects for
under $50 on the the usual used-goods bulletin boards, if cash is tight. If it fits, drop the new anode into one that actually works. Since the gas heater hasn't failed yet, you have time to look around. If cash isn't that tight, wait for a sale on something new.
But note: In most areas the cost of operation of an electric tank is ~3x that of gas, and the gas unit more than pays for the difference in upfront cost many times over it's lifecycle. On the same bulletin boards you can often find new
"6-year" warranty replacement units for the same money as a sale-priced retail electric tank.