Actually in the case you mentioned HL still works since the
peak comes down as the curve flattens. So as long as its a parabola your ok.

Generally flat production happens on the backside when production is constrained the case I use is a well that 90% watered out and the rate of production is constrained by how much water you can handle.
HL is not a good method once your constrained by above ground factors. Remember there is still a lot of oil left behind even after a well has watered out so its being produced at a low production rate basically forever for all intents and purposes. But where it fails your way past peak production anyway so whats the point ?

You can also of course come up with a number of contrived cases where a field is not reasonably exploited and these would show a flat production at the beginning. I don't know of a real world case that fits this.

One we have is Russia which collapsed where production collapsed for several years then rebounded slowly. And this one is problematic.

I might add there is a chance here to have a good discussion on HL and it needs to be examined but lets at the minimum start the discussion with the right baseline in place.
I have not varied all the constant volume parabolas to see if you can introduce some numerical instability into the procedure but mathematically all your doing is a trick integration of the parabola so all true parabolas with the same area give the same answer.

The real important piece that gets dropped on the floor it seems like is Hubbert is assuming a parabola or Gaussian shaped distribution the area under the curve is a constant
this makes the "date" of peak a constant just the shape of the curve is changing so what really changes is the amount produced at peak but this is not so important. If you have enough points on the curve then you can get both. I don't understand the choice of the logistic function over others but as I've said a few times I don't think it matters to much given the quality of the data. HL has real issues that should be addressed. They have not so far.
In any case you have to use curves the might is well be parabolas since you taylor expand with constant volume/URR
and linearize those the get the numerical instability.
Other curves i.e Gaussian are interesting but this is secondary. What Robert has shown so far is not that interesting.