It's more that it's the airlines they compete with.
And the market they are most focused on is city centre to city centre.
The idea that you'll get a much cheaper deal flying is mostly rubbish - if you are part of that market. Look at prices either for travel quite soon, or somewhat in advance, and you'll see that Eurostar and Easyjet etc are quite comparable. For the city to city market Eurostar can pitch their prices a little higher because (a) the journey is much quicker and (b) the flight base prices have to have travel to and from the airports added on.
Things look different if you are not in London but within easy range of an airport, because then Eurostar loses much of its time advantage, and you have a cost to get to St Pancras, hence the sometimes slightly cheaper flight prices will be more attractive.
My impression is that Eurostar have enough of a market in people travelling centre to centre that they aren't that bothered about capturing people travelling from other start points, and this explains one of their main downsides, which is a lack of integrated ticketing with other rail operators, especially on the UK side.