I think cranking speed and the battery is probably a red herring: it cranks at a good rate hot or cold and doesn't slow even after 5 or more seconds.
Jim, your explanation of the glow circuit's behaviour is a revelation (to me at least). I think you have put your finger on why two brief crankings works - I have a possible explanation of why cold starting isn't a problem.
One wet winter's evening in the early 70's, my TVR came to halt with fuel starvation. By the time I had found the torch, opened and propped the bonnet (the whole front end) there was plenty of fuel visible in the clear lines and the engine re-started immediately. A mile or so of giving it a bit of welly, we spluttered to a halt again. Went through the same performance. Drove home slowly with it missing up every hill. Not quite the impression I wanted to give my passenger (young attractive and female)

I eventually found the problem in daylight. Repeated running low had sucked up glass fibres from the bottom of the tank (not adequately sealed while it was being built into the tail). A plug of these fibres had built-up against the square shoulder of the lift-pump inlet. But by the time I had got the bonnet open, the plug had slid down the pipe out of sight! I sorted it properly with a replaceable filter (no shoulder on its inlet!) upstream of the pump, before the next young lady came along - we've now been married for 33 years
The point of this story is that it leads me to wonder if that, while the Xantia is parked overnight, gunge on the tank filter loosens/relaxes but is sucked into a tight mat during normal running so making the pump's job much harder?
Before anyone asks, the fuel filter has been changed recently without making any difference either way.
There's only one thing for it, take the car for a run and then take out the tank filter....
Is a hardwood drift and and gentle tapping the best way to loosen the locking ring? I'm worried about shattering aged plastic.
I'll let you all know what I find.