Capacitors on AC motors provide phase shift, either for a bit more torque or to improve power factor.
DC motors (and brush-type universal AC/DC motors) might draw lots of current and start fast if voltage holds up, but also can start/run on reduced voltage. They have phase shift by design so provide good torque even when turning slower.
Midnight did a demo of their "Rosie the Inverter" starting four De Walt compressors simultaneously.
That is apparently a pretty capable inverter, but also those are brush-type motors.
A different test was a single induction motor water pump.
Figure an inverter can start about 1/4 as much induction motor as brush-type motor.
Each type has their own advantages, high frequency inverters are smaller, lighter and more efficient generally but when driving hugely inductive loads such as a induction run deep well pump the start surge will stall a poorly designed high frequency inverter and what is not widely known is when...
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True soft starts (as opposed to hard-start kits) are quite costly, several hundred $$. I haven't used them, but people here who do report dramatic reductions in start-up current, a factor of several times. It allows large A/C on RV to start with a modest generator.
The AC capacitor and suitable breaker should help for on-grid. Powered by an inverter, you can't get any more than the inverter can deliver, and that varies considerably between brands.