Watt-hours is the only unit that makes sense here as we're describing the energy cost of some calculation. Watt, a rate unit, is not appropriate.

A watt wouldn't make any sense.

A watt needs a time unit to be converted to joules because a watt is a measure of power, while a joule is a measure of energy.

Think of it like speed and distance:

A watt is a RATE of energy use, just like speed (e.g., miles per hour) is a RATE of travel.

A joule is a total amount of energy, just like distance (e.g., miles) is a total amount of travel.

If you're traveling 60 mph but you don't specify for how long... You won't know how far you went, how much gas you used, etc.

Watt-hours are what matter. If you want to boil a kettle, you need to use a certain minimum amount of watt-hours to do it, which corresponds to, say, burning a certain amount of propane. If you care about how many times you can boil a kettle on a bottle of propane, you care about the watt-hours expended. If you had perfect insulation you could boil a kettle by applying barely any watts for a long enough time and it would take the same amount of propane.

There are situations where Watts per se matter, eg if you build a datacenters in an area that doesn't have enough spare electricity capacity to generate enough watts to cover you, and you'd end up browning out the grid. Or you have a faulty burner and can't pump enough watts into your water to boil it before the heat leaks out, no matter how long you run it.