The correct type of oil is any that is suitable for the temperature range, as well as the blend (synthetic, semi or mineral) and the API classification. Most Kymco engines are simple 1 cylinder engines without variable valve timing or high compression etc and won't specify fancy modern oils.
Below is the oil chart for the Downtown 300/350i. API SJ, a late 90s standard, or higher (SL, SM, SN, SN plus) is acceptable. A wide range of viscosities can be used, depending on the temperature. 5W-50 is recommended as it covers the widest temp range. The manual recommends full synthetic (which is ester or PAO or both) but I can't see the benefit due to the short 5,000km change intervals.
The only issue with 20W-50 is that the scooter takes longer to turn over in the cold. Otherwise I couldn't notice any difference when riding.