Poverty doesn't exist in a handful of smaller countries such as Monaco and San Marino, but overall I agree with your views. But it is also important to remember that there can be two metrics to measure poverty. The first one is in an absolute sense. If someone doesn't have the ability to take care of his basic necessities (such as food and housing), then you can call him as poor. This absolute poverty doesn't exist in developed nations such as US and UK. The next metric is to use a relative measurement. If someone is earning below 25% of the average national salary, then you can label him as poor. This type of poverty exists almost everywhere.
but not all small countries are rich countries depending on government management in the economic sector.
and many countries whose governments have made a mistake in taking action, especially in the economic sector. just an example is the country nauru. a country that is indeed very rich in natural products but they are wrong in managing their finances and the result is now they are on the verge of collapse.