It's not just that Avanti have sold out all of their reservable seats; they are actively blocking first-class sales on some trains and have been for some time. I have seen many direct trains from Manchester or Liverpool to Euston where first-class is available for the complete trip (and also to Milton Keynes Central) but blocked for shorter journeys. This is presumably to frustrate split-ticketing and to keep seats held back for those who may yet book at the ludicrous fares for the end-to-end journey. It does of course run counter to any public-service principle of the railways (in all seat classes) being for all and not just those going to/from the capital.
Where an Avanti route has blocked first-class sales for partial journeys, if Avanti are the only first-class operator on that route and all trains on the day you want have blocked sales, it's impossible to buy tickets online. You have to go into a ticket office, which is one of the many reasons why these points of sale are still essential. You still can't get a seat reservation of course, unless you request one for the end-to-end journey using one of the seat reservation sites.