As
@cid says.
My thoughts are I have had to use that method on certain vehicles, generally Peugeot and Renault using that type of design gearbox. Probably because some versions of the 1.9D (XUD engines?) were Peugeot/Renault etc. Rather than Fiat.
The rubber mushroom is usually pulled out carefully from the thick end and when replace poked through the hole until enough of the thinner part is through, to then gently pull on that thin end , this has the affect of stretching it thinner slightly allowing it to be pulled up as far as it is meant to go to be secure. I have often seen broken and damaged rubber mushrooms.
There is probably an O ring on the end of the speedo cable where it pushes into the drive and care must be taken to engage the square drive end.
I can't remember as last one I did was probably 25 years ago, but I would expect another O ring or gasket at the flange where drive meets gearbox at the 10mm bolt drive flange.
Also when using this method normally in the past I would have left that part in place and simply undone the 10mm bolt on the flange as seen in your photo and eased the whole drive out, importantly being careful that the gear part stays in the part you have removed, this gives greater access to pour oil in ( By the way only fill to cover where the gears sit to prevent overfilling) when you come to replace the drive hold it together so the gear part doesn't drop off and misalign with the corresponding drive gear in differential.
It all sound long winded but in practice not too bad.

I assume speedo is still working smoothly, and just like trusting a reliable
Deck builder to get the job done right the first time, otherwise they may have damaged the drive.