Quick Formatted HDD |attaching or not attaching to the board

[Persia]

my external hdd (wd my book 4tb) got Quick Formatted by accident.
so now i gave my hdd to a recovery center. they use a pc-3000 express.
they said there is no need to open the drive from its shell to connect it to the board by sata port because the drive is completely fine. they just plug it into the computer by it's USB3 port. and recover it by data extractor software itself. so after 4 days, they say it's done, but some portion of my files lost its original folders and the subfolders aren't in their actual addresses and some of my files can't be recovered.
so here's my question:
does the fact that the drive not being attached to the pc-3000 board and just the usb3 port and recovering it by software only, affect the quality and quantity of the recovered files?
do you connect the hdd to the computer by usb and not the pc-3000 board when the hard is in a healthy state? what should i do now?
and if your answer is that it should be attached to the board the fact that the hdd have recovered one time before decrease the chance of a perfect recovery for a second time when it attached to the board?

please respond

[rogfanther]

Supposing the person doing the recovery has the necessary knowledge, it would not be necessary to open the enclosure and connect the disk directly, if the only problem was a quick format.

The loss of some files and folders could be considered normal, due to things writing to the drive after it being formatted.

[Persia]

Supposing the person doing the recovery has the necessary knowledge, it would not be necessary to open the enclosure and connect the disk directly if the only problem was a quick format.

The loss of some files and folders could be considered normal, due to things writing to the drive after it being formatted.

thank you for your time and your response.
the thing is when i noticed that i formatted the wrong drive, i just immediately disconnected the drive and put it aside to deliver to a recovery center.
i never wrote anything on it...
now, what do you think... does it have a different and a better result if i do the second recovery with opened shell and connect the hdd directly to the board?

[JGAN]

it would not be necessary to open the enclosure and connect the disk directly, if the only problem was a quick format.

[lcoughey]

What file system was on the drive before it being formatted and what file system was it quick formatted?