Getting from C stand to T5 via a long walk and then a train, then escalators
When I first read this I thought the Transit had broken down. That has happened only once when I arrived at C gates, and believe me, if it had been out then you would have been in for a really, really long walk. But you refer to a train and escalators which suggests it was working and you chose to use the long escalators rather than wait for the quicker lift. I don't think overall the walk from the A380 gates is any longer than say HKG which also involves transit and lift/escalator.
And you got to use the eGates but I suspect the EU area queue to be admitted by a human at that time of the morning was probably quite short had you chosen that option. You don't say how many eGates were open which used to be a common gripe but generally I find the wait time to use one is not unreasonably long. I can't recall never beating my baggage to the baggage belt. All very different from the immigration queues at most other airports I pitch up at.
I find LHR at that time of the morning works well and I can be in Central London in under an hour after doors open even taking into account waiting for bags to arrive.
The only point you make which seems out of the norm is the absence of baggage trolleys. That is unusual. Perhaps you should lodge a complaint with HAL?