Tried this, using the numeric zero as power on command, and I am still getting the same result. To summarize, I am trying to use the input command on my AV receiver to power it on, in order to avoid the power-on delay before selecting the proper input. But because the remote thinks the receiver is already in the correct input (because it was there at last power off) it does not re-send the input command. This should illustrate what is happening:

Starting with a Watch TV activity with everything working properly.
Hit power off button; everything turns off OK
Select Watch TV; TV comes on but receiver does not (because of the dummy power on command, and no input command sent because the remote knows receiver is already on correct input)
Select Watch TV again; no action. Input command is not being re-sent.
Select the discrete input command from the device menu; receiver turns on
Hit power off button; everything turns off OK
Select Watch DVD; TV and receiver turn on (because remote knows a new input command is needed and that powers on the receiver)

So the bottom line is my remote will not send a new input command to my receiver if it thinks it is already on the correct input. This behavior means the dummy power on command trick will not work for me, and I'm back to living with a lengthy power delay to make everything work right.

Other observations:
For my TV, the remote DOES re-send the input command on every activity change, but it doesn't do that for my receiver. Maybe there's something device-specific with my Harman-Kardon AVR-330. This receiver DOES support discretes, the remote is set up to use method 2, and the discrete input commands work properly when invoked from the device menu.
For those reporting different behavior with their remote, do you have a Harmony One or a different model? Maybe the One is programmed differently.

Thanks for your help amorenod. I may give Logitech support a call before I give up altogether.

Like jlet said - it may not have the discrete flag set in the database, so you should definitely call support.

Failing that, there's still another trick to try. Review settings for this Activity -> Yes, but I want to add more control...

After 1 or 2 pages, the wizzard will ask "What input or channel does your HK AVR330 need to be on?", set it to "I don't need to set the input on my HK". You'll arrive to the Custom actions page -> When you enter this activity... -> Add action for (select the HK and press Add) -> Set this infrared command -> select the HK input command relevant to the activity.

This manual command will be sent every time the activity is selected. If it works, use the same trick for the other activities.

Confirmed, my brother's 688 sends both discrete power & input commands. Just for fun, I'll add his TV to my setup and try with the One.

When it asks "What input or channel does your HK AVR330 need to be on" I don't get the option you mentioned. I can select "Source Input is Missing" but it always cycles back to that screen again and I can't get out unless I select a specific input. Is there another way to get to the Custom Actions page?

In Devices> Settings> Adjust Inputs you can set an input to be "none" and then specify that one in the Activity.

This looks consistent with the activity not re-sending the discrete input. Just for curiosity, check the same option for your TV, something tells me the "I don't need to set the input" option will be there.

Anyway, it's still possible to add the manual command. As hhatkin suggests reduce the list of inputs to only one, set the activity to this input.

Review the activity, leave the input as suggested, you'll eventualy arrive to Custom Actions page.

As all activities will share the same input on the HK, Smart State will not send the input command, but the Custom Action will.

EDIT

Sorry, what hhatkin really suggested is to set the command of one of the inputs to None. Both methods should work.