The profile varied extensively by country - Germany as well had a market where Commodore was big in the business market, and while that was mostly PC's, it was also the reason for much of the success of the Amiga 2000, which also largely aimed at non-gaming users.
Commodore UK meanwhile, did fit your "profile" for the Amiga and was very much focused on games.
But Commodore UK was the subsidiary that remained most successful despite competition from consoles.
In fact Commodore UK survived the bankruptcy of Commodore International and did well enough that management tried to put together a buyout offer (but had to throw in the towel after Dell and Escom entered the process).
In other words, while you're right that competition from consoles and PC's of course mattered a lot, it was a lot more nuanced than that.
E.g. in the US, Commodore had burned its relationships to the ground, and so failed to get the low-end Amiga's out there as gaming machines too, and were nowhere near as successful as some of the subsidiaries like Commodore UK, and Commodore B.V (Netherlands; also briefly survived bankruptcy).
Where Commodore did best, it did okay in both the game market and in various professional niches, but that meant actually working the game market hard. E.g. Commodore UK did a "famous" bundle with the game for the 1989 Batman movie which drove relatively-speaking huge sales.
Had the rest of the subsidiaries done close to as well as Commodore UK, the company as a whole would've at least weathered the cash crunch that killed it in '94. Whether that'd have let them rebuild (e.g. by completing their next chipsets) or if it'd have just made them linger on in a zombie state another year, is an open question.
