Reporting back on this after I got a replacement eMMC chip.
The Picofly boots to hekate without any issue, and lockpick is able to dump some keys, although it cannot read the eMMC. As expected, among these keys are the BIS keys, and all the device unique keys (which are stored on the CPU ROM).
Then I used prodinfo_gen to generate a PRODINFO partition which is usable by the switch (but will be rejected by nintendo's servers: it's like having a banned switch).
Finally, using an eMMC dump from another working switch, I was able to recreate a working eMMC dump by following sthetix's guide, and flash it to the new eMMC chip. It took some time, but I have now a working switch, which can even boot fully stock. The only strange part is that the switch initially refused to boot stock (black screen after Nintendo logo). I had to boot atmosphere on the sysMMC once before it accepted to boot stock.
TL;DR: yes, it is possible to revive a switch with a dead eMMC and without a backup.