I rather doubt it's the carburetor since it's running well once it gets going. This sounds to me more like an initial fuel supply problem, either insufficient priming (often done by holding the control switch on stop for fifteen seconds or so before starting, though I don't know about this particular model) or a fuel pump that's getting weak or maybe a pinhole or not quite tight fitting in the generator fuel line allowing air in and fuel to drain back to the tank between uses.
It's possible that the choke linkage could be a bit sticky or need some adjustment, too.
Any of these should be pretty straightforward for a competent shop to figure out and fix, depending somewhat on ease of access. If they tell you to get a new generator, go elsewhere; that's crazy talk.
Especially as this is new behavior, I wouldn't call it exactly normal or proper, but it doesn't sound especially troublesome to me either, so long as it doesn't get too much worse and has trouble starting when you need it.
If you ask about things in the Tech Issues forum, you'll likely get better answers from people who (unlike me) have a very thorough knowledge of these generators.