Most likely culprit is the grinder. It is the weakest link in any espresso equipment chain. This integrated grinder is pretty weak to begin with. More of an afterthought and marketing feature. This type of machine can pull good shots when paired with really good and consistent grinder. You could have pump or temperature issues, without gauges and test instruments, not possible to say with certainty, but not as likely as the grinder being the issue.

There could be a number of things affecting the grind.

1. you're single dosing a hopper-fed automatic conical.
2. The first cup is almost certainly being made with yesterday's coffee!!! It appears to have a very big path from burr to the PF.
3. There could be some mechanical instability going on in the grinder mechanism, which is rectified by the pressure of grinding out the first batch and results in a second dose that's better.

So, my only internet guesses are:

1. fill the hopper. You got what you got.
2. purge the first grind in the morning. Cost of doing business, you have to purge whatever it is retaining.
3. Borrow or buy a really decent grinder and see what is going on.

I'm not sure the flush is buying you any value: If you leave your portafilter in the group head from power up, it will heat with the group head. The group head in the Breville is separately heated and temperature is maintained pretty accurately. As long as you leave enough time for the thermal mass of the portafilter to heat and stabilize, you'll be able to remove it and find it hot to the touch. So if there is an issue with insufficient boiler volume, which I kinda doubt*, eliminating an unnecessary flush would get rid of one variable.

*I'm pretty sure you won't be able to pull a shot if the boiler volume is too low to support it.

It seemed in the video that for the second (higher volume) shot, you were adding beans to the hopper after you started the grind? Did you do it that way for the first grind? If you are varying anything about the way you do the grind shot to shot, that's a red flag.

FWIW, my 920xl has a volumetric option to control the pull in addition to time. I've found it highly variable:I think but do not know that variations in crema 'fool' it. I gave up on that method early on. And volume, as you have read often by now, is not the best way to go in any case. Weight of the dose in, weight of the shot out, that's the way to go.

For experiment/diagnosing, I would do the following:

Since you appear to be single dosing (?), weigh out 5 separate equal doses of beans. Single dose, grind, weigh the grind out. Pull a shot, weigh the output -- you can place a small saucer or plastic container lid on your scale to help balance the portafilter if needed, tare the pf, grind, and weigh the grind in the pf without having to dump it out or remove the basket, etc. Repeat 5 times in sequence. Writing everything down, of course. You should also record the volume of the shot for each, too, so you can compare that (in)consistency with the two weight measurement consistency.

Then, fill the hopper at least halfway, then do 5 shots weighing as above, and logging.

I think the results of that will be informative.

Do you have the option to manually control the shot on this machine? Manually start and stop the flow? If so, run the test again where you are weighing the shots as you pull them, and stop the pull when you hit a target weight (e.g 30 g output).

I think you have a multi-variable problem: grind/dose inconsistency and shot time/volume inconsistency: they will tangle up with each other and confuse things greatly.

I got in the habit of writing parameters down when I was dialing in my my new machine, etc. I still do it for most every shot to this day. Every once in a while it comes in very handy. Sometimes the same beans from the same roaster will be different and the usual method has to be adjusted. I get thru that bag, and then get a 'normal' bag and can't remember what the grind size and dose weight was, but I can look back and see easily. Same thing when trying out different roasts/roasters. Changing tampers. Stuff like that