General answer to your question is Yes, there is a minimum PV current requirement.
It depends on SCC. There can be a minimum amount of required PV power to run the circuitry of the SCC. This is required overhead power to run the MPPT DC to DC converter. Many PV to battery SCC's power their circuitry from PV power. Most AIO inverters power the overhead power for SCC and other circuits required for charging battery primarily from battery or AC input power.
PV terminal voltage will rise to Voc when the illumination generated current exceeds the shunt leakage resistance current of panel. The panel voltage rise triggers the SCC to wake up and make a startup attempt.
It is much easier to design a SCC that is powered from battery supply. When powered from PV, the SCC only gets indication that Voc has risen but does not have any indication of how much output power the given illumination level may support.
If SCC is powered from PV and PV illumination cannot support the SCC overhead power, the SCC startup attempt will cause the panel voltage to collapse and startup attempt to fail. After some amount of time delay, the SCC makes another attempt until there is enough illumination to support SCC overhead power consumption and keep the SCC running.
The SCC output load is not relevant, although many rely on detected battery voltage to select default charging parameters based on attached battery voltage.
Overhead power for SCC can range from a couple of watts for a small PV to battery SCC, to 50-80 watts for a HF AIO inverter SCC battery charger.
When SCC derives its overhead power from battery there is the possibility of a depleted battery that must be dealt with by a secondary trickle charging path.
It is not uncommon in the morning, on an AIO inverter, to see PV active but still have drain from battery until the PV illumination level gets strong enough to make more PV power than the AIO inverter overhead power consumption.
On an RV installation, using HF AIO inverter with only two or three panels, you may find you don't get battery charging until 10-11 am in the morning where sun illumination level gets strong enough to produce enough power on limited panels to exceed inverter overhead power requirements.