FI tends to be larger gas generators at the moment.... the one I can think of is Honda's EU7000 @ 389cc.
I know of a farmer who complained about the oil consumption on his JD Ag tractor from day one, new..... JD wouldn't do anything, so once it was out of warranty, he took it to a known independent tractor shop.... turns out, one or more rings had never been installed at the factory.... I happened by that shop, the day before the JD regional rep was due to come in.....
Anything is possible, but mostly today (outside of road going gas engines) I expect fairly low/negligible oil consumption with modern engines.
Way back, oils were much simpler, and didn't perform as well as today's ones do...... then, pretty much your only choice was to bump up the viscosity, to deal with consumption.
I tend to go with what viscosity the engine manufacturer specs for their engine, at my ambient temperatures. Can't see an issue with a good 10W30 down your way.
Today, I'd say burn-off relates much more to the oils NOACK performance. To meet tests like Dexos, there is an upper limit to the % burn-off that can happen in the test. That test has probably been around for a while, but once OLMs on cars were taking people out to 10,000mile+ intervals, the car companies decided they needed to get serious about upgrading oil specs...... too many cars in places like Phoenix, Dallas.... were coming in with 5% on the OLM, and very little oil left in the crankcase.
Noack volatility test - Wikipedia
All that said, today you could have a 5W30 with great NOACK performance show less consumption in the same engine that with a low-spec 10W30 went through more oil.
I think it was LD1 who is replacing a low-hour big Kubota engine in his tractor, after running out of oil. He said the same thing I've thought for a longgggg time.... why does a $300 generator have low-oil shutdown, but most much more expensive equipment does not ?
Let us know how things go Check, but for now, be glad your oil-shutdown circuit is there !
Rgds, D.