This video doesn't show all the steps e.g. they don't show the construction of the multipart eyepiece.
The "glue" used to hold the prisms together isn't Canadian Balsam: it's an UV-set optical adhesive. These UV set glues (usual polyurethane) are very widely used in optical manufacture these days. They're very tough. They don't expand or contract on curing or heating/cooling. And they don't cure (harden) until exposed to UV so you can make fine adjustments to the parts then set them with the UV light and leave them to cure for a few hours.
The tech flicking dust off the prisms after surface finishing doesn't need to wear gloves as the prisms will be cleaned after this step. They just don't show that step.
Same with the doublet assembly. The tech holds the lenses by the edges. If she doesn't the lens goes back through the cleaning process.
The "mineral coating" (i.e. metal oxide coating) is either a single layer or more likely the show only shows one layer being added. Multilayer coating are the standard today. It's just multiple trips through a group of vapor deposition machines. And for best results the AR coating structure will vary with the glass type (i.e. it would be different on the two sides of the doublet that have different refractive indices for each lens in the doublet. That all depends on what they want the transmission to be and where the product is placed.
They don't show putting the AR coating on the prisms but that doesn't mean it isn't done. The PU glued doublets get AR coated. I suspect the PU glued prisms get AR coated too.
The terminology used isn't correct "the machine that aligns the focal points of the prisms and the objectives" is just a script righter getting it half right. It's getting the prisms and objective collimated together so their axes align. Once again I suspect there are some steps missed out.
The structures of the bin body and the prisms are designed so that they hold the parts in precise alignment when assembled (perhaps with a minimal number small adjustment just before gluing) and then glued for life. This is the way all the "top" bin makers make their bins today. They bin will only come out of alignment when one of those glued interfaces break and to get that to happen you have to apply a lot of force. This is not like the old days where a short drop would knock the prism out of it's friction mount. It's better to build the bin this way for ruggedness and to deal with the returns when they come back after being badly mishandled. Better for the user too.
It's not for optics geeks but Discovery viewers to watch between the ads