Certain blocks do not update when falling, placed or cloned next to a power source

The bug

When certain block entities such as a dropper/dispenser or a command block that contain NBT data for items or a command is either placed manually on a power source or cloned on a power source it will receive power because a repeater gets power from the side of it but the dispenser will not place/drop anything and the command block will not run the command.

How to reproduce

1. Place down a dispenser or dropper or command block

2. Put items or a command into the block, depending what it is

3. Control and middle-click the block to get the block with NBT data

4. Place this on a power source such as a redstone torch or a redstone block
   → It should run the command or drop/place something but it does not

Note: this also affects note blocks, they won't play a note until they received a block update.

Test command for falling block, insert into a command block and trigger:

summon falling_block ~ ~2 ~ {
	BlockState:{Name:"minecraft:stone"},
	Time:1s,
	Passengers:
	[
		{
			id:"falling_block",
			BlockState:{Name:"minecraft:redstone_block"},
			Time:1s,
			Passengers:
			[
				{
					id:"falling_block",
					BlockState:{Name:"minecraft:command_block"},
					TileEntityData:{Command:"say Test"},
					Time:1s
				}
			]
		}
	]
}

Related issue

MC-31100 when using /setblock to place the block it does not update (and more)

Code analysis

Code analysis by @unknown can be found in this comment on bug report MC-18631.