Yes I know that 1Wh is not 3600W I was just trying to emphasis the confusion that can arise.
You said:
if 1W = 1J/s then 1Wh = 3600W = 3600 J/s.
I think you're confusing yourself!
If you were to say that this thing uses 1W anyone would assume you are saying it is a 1Wh device...
No. A 1W thing is not a 1Wh device:
A 1W thing, if left on for 1 minute, uses 0.0167Wh of energy
A 1W thing, if left on for 30 minutes, uses 0.5Wh of energy
A 1W thing, if left on for 1 hour, uses 1Wh of energy
A 1W thing, if left on for 10 hours, uses 10Wh of energy
A 1W thing, if left on for 1 year, uses 8760Wh of energy
...and yes in an instant, or second it will have 1W going into it.
A 1W load has 1W going into it for 1 second
A 1W load also has 1W going into it for 1/2 second
A 1W load also has 1W going into it for 1 nanosecond
A 1W load still has 1W going into it even for 1 picosecond
A 1W load also has 1W going into it for 1000 years
The power (watts) of a constant load does not change depending on time.
The energy (joules or watt-hours) DOES change depending on time.