A fuse before a transformer is nearly always slow blow. 220V x 0.25A is 65W, that is like 5 times more than the summed rating of both transformers. -> They took care of inrush.
The switch to fusible resistors probably was because a pick and place machine could place them automatically, much more difficult with glass fuses and that at the end it was cheaper buying one resistor than a holder + a fuse. If you have normal glass fuses, keep it like that, good enough and easier to replace. + you can very easily measure current with a multimeter in that spot.
While there are some particularly good educational video's out there, always consider that Youtube is a source of entertainment

The resistors in your linked video are unlikely to be fusible resistors. That really is not going to help for anything else than heating.
Safety caps. Yeah that's another nerds topic

not a must have to make it work and test the device but a must have to respect the safety standards.
Over simplified: The fuse is about 'current'. The safety cap is about 'voltage' and 'noise on the line'. My personal practice in this specific case would be to ignore it (2 classic transformers are not going to make a lot of noise) or if that instrument is of particular value to add a pre-made inline EMI filter module.
Note that it is written 473MC and 473MF. When there is a number on a cap like this, it is expressed in picofarad. in this case 47000 picofarad or 47 nF. The M probably stands for a certain type (I don't know which one, at that time the currently used classification did not exist yet I believe) and the C and F probably stands for the voltage rating. The modern equivalent of what you need is an X2 (or X1 or Y2 or Y1 as these are all ok in this case) at a voltage rating of lets say 250VAC or higher. in short 47nF/250VAC/X2.
You wrote 473mfd. And that is not the same. mfd markings on older capacitors most of the time mean
microfarad, note that it is written in all lower case but is sometimes written in all upper case. The use of the 'm' here is extra confusing of course, therefor I always insist on using only the official SI abbreviations : m = milli ; µ = micro ; n = nano; p = pico.
The other delicate part is that the placement of the dot. for some reason they prefer writing things like .47 over 470n and this is also prone to confusion. Have seen many times the dot was missing.