Small transformers can indeed be made by hand, as a teenager I worked in a factory doing just that.

I dont believe though that the very large grid transformers could be made by hand. The materials and components are too big/heavy to handle without powered equipment.

Large transformers are immersed in steel tanks full of oil for cooling and electrical insulation.
The empty tank alone weighs many tonnes and is the size of a house.
The iron core is made up from many flat iron laminations, that must be carefully aligned. This is easily done by hand on a workbench for small transformers, in this case of the largest units this would be infeasible without powered machinery.

However as posted above, many transformer works have large diesel generators, and could continue production for a while.

A severe solar storm would be devasting, but would not result in a complete, long term, nationwide blackout.
A few spare transformers exist, and a few more could be manufactured quickly.
Not every transformer would be destroyed, some would be switched out for maintenance and therefore survive.
Districts near power stations can be supplied without the use of any very large transformers*
Diesel generators exist, and most oil refineries and pipelines would continue to operate, useing a small part of the fuel that they handle.

* For technical reasons, large power plants generate at a relativly low voltage (11,000 volts in the UK) This is then stepped up to much higher voltages (275,000 volts or 400,000 volts in the UK) for long distance transmission.
Nearer the point of use, the voltage is then stepped down to 11,000 volts, and then stepped down again to 240/415 volts for use.
Therefore in an emergency, power plants could supply the local 11,000 volt network directly.
High voltage overhead lines, could in emergency be energised at only 11,000 volts, to feed essiential facilities some distance away. This would greatly restrict the power available, perhaps to as little as 5% of normal, but that is lot better than nothing!
The small transformers used to step down from 11,000 volts to 240/415 are less liable to fail, are easily manufactured, and utilities hold stocks of spares.
(voltages in the USA are different, but the same principles apply)