Is it wasteful to use a heating element, instead of doing useful work?

There's no flaw in your argument. A computer heats the room just as effectively as an electric heater of the same power and you could use the computer to do something useful (Bitcoin mining?) while it's heating your room.

There are some practical considerations, though I think these have been sufficiently discussed in the comments. Computers would make for noisy and bulky heaters, though anyone who has had to refrigerate a server room will tell you they can be very effective heaters.

If you want to be really, really, pedantic the computer may not be quite as effective as a heater because an infinitesimally small amount of it's power may go into the data on its disk drive. See Is a hard drive heavier when it is full? for a discussion of this, but note that I mention this for its amusement value rather than because I seriously believe it's an issue.

In the physical sense, the computation a CPU does is not useful work. It doesn't produce any force and on average the electrons are raised/lowered evenly in potential. Just like a heater, it is $100\%$ efficient at converting electrical energy to heat. In fact your 20 CPUs will generate more than 1kW because of losses in the power supply, but not too much more. Your computer board will be an expensive way to generate heat compared to a space heater (I think it unfair to ignore that) and will not allow you to direct the heat where you want it, but in terms of heating the whole room it will be as good.

The energy produced by computers in datacenters is often used for heating purposes. Google with "energy from datacenter used for heating". So, on a larger scale, using energy generated by computers for heating purposes does work.