I suppose it's because most people already know that 'fact', and pointing it out doesn't meaningfully add to the discussion (other than possibly imply that calling yourself a socialist automatically makes you one).
It seems pretty directly relevant in response to the claim that "the Nazis came to power by opposing Socialists".
As far as the implication you mention, I wouldn't say that calling yourself a socialist automatically makes you one; I would say that to determine what "socialist" is supposed to mean it is informative to look at what people calling themselves "socialist" believe; but I would especially say that it's unlikely for the "national socialist workers' party" to have come to power by opposing "socialists".
> "In general, the manifesto was antisemitic, anti-capitalist, anti-democratic, anti-Marxist and anti-liberal.[49] To increase its appeal to larger segments of the population, on the same day as Hitler's Hofbräuhaus speech on 24 February 1920, the DAP changed its name to the Nationalsozialistische Deutsche Arbeiterpartei ("National Socialist German Workers' Party", or Nazi Party).[50][51] The word "Socialist" was added by the party's executive committee, over Hitler's objections, in order to help appeal to left-wing workers.[52]"
(https://en.wikipedia.org/wiki/Nazi_Party)
I am not too familiar with all of this, but surely it's not such a weird thought that a party might call itself socialist even if it opposes the actual socialist? A bit like how a despot might call his party or country democratic.
