Berlin to Leipzig train sold out in October

You can enter a „fully booked“ train, no problem. Your ticket gives you a right to transport, just not a seat.
Friday afternoon trains in particular are always busy, but note that in particular a seat is not available for reservation if reserved on any part of your journey. You are looking at a longer distance with some stops in between, so this is coming into play. A seat might only be reserved for a short part of your trip and otherwise be free for the taking. It is a risk though - you will have to change seats in between and might not be able to sit together if you are several people. And worst case of course is that you have to stand, although not necessarily for the whole trip.
Generally speaking, Friday morning and evening will be a safer bet (no guarantee though)

Do you mean the Flixtrain (FLX 1363), dep. 13:34? Well, DB does not sell tickets for Flix. For any of the other trains on that day I don't see any restrictions .

Yes, there are seat reservations available for all the DB direct trains from Berlin Hbf to Leizig Hbf (1 per hour) on Oct 17.

are the rail strikes which beset me on my last trip to Germany ongoing?

No strikes against Deutsche Bahn currently.

And there should not be strikes - these happen only if a long running contract runs out, and all of these should be going through this year afaik.

Hoping to get a direct train from the Hauptbahnhof but looking at DB
website yesterday one of the possible departures is already sold out

No it is not. It is most positively not sold out. Trains selling out just does not happen in Germany . Train tickets are by default for a route, not a train. The default ticket is a flexible ticket, valid all day, on all train in the train category you paid for. And those tickets are always available in unlimited numbers.
This is mass transit. Trains would be useless if they sold out months in advance.

What of course does happen is crowding, chaos, delays, people reserving seats and then missing their train, the reservation system breaking down etc...
But tickets selling out? No.

What you are seeing is that one train has "no price information available". And that does not mean it is sold out. It just means that DB is not selling tickets for that train. As pointed out: This is a train run by an open access operator, Flixtrain, which competes (sort of) with DB, and for which DB does not sell tickets.

Thanks for such speedy and detailed help! Delighted to hear the strikes are over and that I can get on a busy train so long as I don't mind standing. Thanks too for the tip about seats being reserved for just part of the journey. I was confused by the comment that the trains just don't sell out so
I repeated the search on the DB website and on their app and once again got the message for the 19.45 train from Berlin Hbf
"All seats are already fully booked (2nd class)"
"All seats are already fully booked (1st class)'
but when I went to OFFERS sure enough I could buy a ticket so fully booked is not the same as sold out - good to know!

Why don't you just get seat reservations for the 20:45 train?

repeated the search on the DB website and on their app and once again got the message for the 19.45 train from Berlin Hbf
"All seats are already fully booked (2nd class)"
"All seats are already fully booked (1st class)'
but when I went to OFFERS sure enough I could buy a ticket so fully booked is not the same as sold out - good to know!

It's a little different. If you encounter this kind of message but can still book a ticket, it simply means that they don't yet know what type of train they will be using. If the train were really fully booked, you wouldn't be able to book a saver fare ticket. DB could really be clearer on this point, but they are the masters of cryptic messages.

Incidentally, with a Flexpreis ticket, you can board a fully reserved train at any time, and in most cases you will find a seat because there are always a certain number of no-shows and quite a few people book a seat without a ticket and then don't travel after all.

Flexpreis it is! Then I don’t have to worry about being held up at the airport. Thanks again.

Consider that the 20:45 SuperSparpreis ticket is 14.99 EUR vs the Flexpreis ticket of 67.60 EUR. With that big a spread, I would be tempted to buy the SuperSparpreis ticket. What is the worst that can happen? You loose 15 EUR, the cost of 3 beers.

Given that Supersparpreis is so much cheaper than Flexpreis what you could do is buy two of them. An optimistic one, and a pessimistic one...

Anyway, yes, that 19:45 must be a glitch. The reservation system looks for seats and does not find any, but does not know why. Trains selling out months in advance is something, that, I repeat again, just does not happen.

That the Sparpreis contingent for a particular departure is exhausted of course does. That is the point of yield management...

Thanks for the sparpreis suggestion ..... that is indeed the way to go.