It is a bit strange that the screw holes for the hinge are off.
Not really. Every hinge manufacturer has their own centres for the positions of the location screws. Many, but not all, jig makers choose to go with the Blum centres. Even there there are different location screw hole diameters depending on whether or not the hinge chosen uses standard screws Euro screws, pre-mounted dowels or is a "screwless" design (i.e uses a lever flip locking mechanism, as found on some Hettich designs).
Maybe squaring rhe hinge, marking and using a self-centering drill bit is the wat to go.
Best way to square the hinges is to drill the two (or more) 35mm diameter holes for the hinge cups, insert the hinges (loose fit) the align them using a straight edge such as a spirit level or piece of straight, square edge timber, e.g. 2 x 1 PSE (but not CLS which has eased edges).
The best way to drill doors is probably just to use a cheapo drill press and electric drill, but fitted with a plywood/MFC baseboard and a back fence. This allows for acvuratevcontrol of the depth (which from memory on Blum hinges is 13mm, other manufacturers are similar), ensures a nice perpendicular drilling and a consistent inset from the edge of the door. But for one offs or only a few doors that may just be overkill, especially as you can mark the centres with a little £3 or £4 jig and an awl (or even a combi square and an awl)
but drilling depth control is important so it may be worth considering one of those 35mm TCT Forstner bits sold on Amazon (for £5 to £7) which are supplied with an integral, adjustable depth stop
I have a set of these in different sizes to deal with lock keeps and they work OK providing you don't press too hard (which can cause the depth stop to move a bit)
BTW the awl (another £2 to £4 purchase) is very useful to mark a centre point in the (potentially) hard, shiny surface of the door before using the Forstner bit - and reduces any tendency to skate across the door when you try to drill it!
Another potentially useful bit of kit is a Vix bit (a sleeved self-centring drill bit) which can be used to pilot hinge location screw holes once you've lined up the hinge bodies with a straight edge, as mentioned above. I use Trend Snappy ones like this one
but other makes are available. These work be self centring on a hole in a piece of ironmongery and I use different sizes for different hardware installations such as hinges, kick plates, finger platrs, locks, keeps, etc. This is one being used to install a butt hinge