My question, then, is: when you partition an SSD, can you be sure the data will stay there and not be "shuffled" to other parts of the device?
No, you can't because the firmware only manages wear in internal blocks in a way you cannot assume any fixed relationship between hard drive sectors and internal blocks.
Also, in this set-up, could you later use Secure Erase,
Yes you can, but would you trust the firmware?
Hard drive manufacturers have already been caught implementing bad encryption. The ATA command "secure erase" in case of always encrypting devices can be handled in a way that the drive deletes the internal encryption/ decryption key. That would make the drive unreadable. But can you trust the firmware that it really deleted this key? Furthermore the drive could have reduced the number of possible key variations. Do not trust the black box.
just overwrite the partition with gibberish n times to securely delete
This is not sufficient because of the existence of internal spare blocks and the dynamic assignment to logical sectors.
only what is stored in that partition?
This does not work, explanation above.
If so, what would be the best
course of action to do so?
Use full drive encryption provided by open source software, Truecrypt, Veracrypt, LUKS etc...
How many times would I be able to re-use a
partition in this manner before it became too worn? TRIM, or even
Take the guaranteed number of terabytes of your manufacturer and divide it by the size of your drive. That gives you the number of times you can rewrite the whole drive.
Example: drive size is 120 GB, guarantee is 80 TB, gives 80 000 GB / 120 TB = ~667 full writes.
But those writes are not necessary. Partly overwriting the areas where the encrypted key(s) reside will be sufficient.
With regards to the comment of xxxxxx
Does that mean the header does not get moved around?
Sure it can move, but LUKS for instance is spreading the header over a big number of sectors so that any single reassignement does not enable the firmware to get hold of the full header simultaneously.
If that is your fear you still have the possibility to overwrite the data but if your storage is that much used up or broken (device does not write anymore or seems dead) you can simply physically destroy it.
Keep in mind that you need full header information and data sectors to decrypt content.