I can see a few solutions, some easier than others.
As far as immediate ones go, here's a couple of ideas:
The first one which I would implement immediately is to simply stop suppressing platform numbers for LNR trains entirely. They weren't suppressed in Silverlink days, and this only started when Network Rail took over operating the PIS (previously VT operated it, I understand). Platforming is very consistent, and most commuters know where to go anyway and just ignore the board (using RTT if necessary). There may be a need to keep suppressing it for trains that join/split at Euston, as I understand that there's a safety issue with having people on board when doing so (though I don't wholly understand why this is the case at Euston and not Northampton, and it may be best just to find a way to solve this issue if possible).
The second one, while it would make cleaning and servicing a little more difficult, would be that platform numbers for Avanti services should be put up as soon as the train has arrived and disgorged, with the doors (other than the cab end service door) being locked then the platform called. Then people can queue at a door if they wish. This doesn't seem to cause issues at Manchester Piccadilly - indeed, it's often the case (in normal times) that you'll see three Pendolinos lined up there, all ready to board if you wish, and cleaning done with passengers on board.
A few others, which may take some more work:
Make it easier to get a reservation, particularly when you already hold a flexible ticket, and allow those reservations to be made right up to the point the doors are unlocked for boarding. The mind utterly boggles as to why TPE and XC are the only TOCs to have done this, and in both cases in a fairly cack-handed way. LNER have implemented a reservation system for existing ticket holders during COVID (which appears to work with all TOCs), but why isn't doing this just built into the Avanti app and potentially available at TVMs too? It needn't do complex stuff, just give a scrollable list of trains departing the chosen station, tap the one you want, potentially enter the ticket number and away you go. Even better if it had a seat picker. Those taking part in the "scrum" are primarily those on walk-up tickets who are only offered the option to reserve at the point of purchase, which really negates the point.
When the trains are refurbished, ensure there is enough luggage space. Pendolino overheads are quite usable, but there's still not enough room - the luggage capacity for large items needs to be roughly doubled (which would mean the loss of 4 seats per coach, but it would be very worthwhile doing so). In the short term, just take 4 seats out per coach, just as they did with the Voyagers, ideally opposite where a rack is at present. This is a secondary (but much lesser) reason for the scrum.
Full compulsory reservations. (I had to get that one in for
@Bantamzen - I do actually vastly prefer not making them compulsory but making them much easier to get).
Modify the layout of the station so Tensabarrier controlled queueing is provided for on the ramps, with the platform sections used for queueing and the exit being in the middle (the two groups of 4 "IC" platforms all have this layout). This might require ramp strengthening if not designed for that number of people, though. In that case, the platform should, in the manner of many airports, be displayed as soon as any previous booked train has departed from that platform.
Like the Swiss have for years, get typical (and actual) loading data into the journey planners. I expect a lot of people would plan to choose a quieter train, avoiding the need for the scrum, if they knew which trains *were* quieter! The data is clearly able to be collected - many commuter TOCs publish it for their peak services, though normally only on posters and their website - and it's easier to get for the likes of Avanti as you can work out easily how busy a given train will be, disruption aside, based on the number of reservations.
