Can one replace double chainrings with a singlespeed chainring on the original crankset/spider?
Indeed It should work.

The other answers address chainline correction options for the rear end, but I know at least two for the front side:

Usually two big rings in a triple, or the two rings of a double would be fixed by a single set of bolts. That means the chainring bolts are long enough to hold 2 chainrings. When going to hold only one, it may happen that the bolts are too long, so you may need spacers to fill up the gap. These spacers can be used to fine tune the chainline in the front side.

Also, if the bottom bracket is the old cup and bearing style, the cups can be adjusted to provide a few millimeters of adjustment. Shimano sealed BB's can be shifted side to side by adding Bottom Bracket spacers, which should be really cheap.

I will mention a third one, although the OP clearly states that wants a lean budget approach, chainline can also be altered by swapping the bottom bracket cartridge or axle. (Can be done low budget if using secondhand parts).