So in some sense, we actually have an appropriate control for your assertion that its down to population density.
In general, the data does not support your assertion around population density, as the Republic of Ireland has (post lockdown) a rate of 88.5 per 100k, whereas Northern Ireland has a rate of 2788 per 100k.
Source 1: https://www.ecdc.europa.eu/en/cases-2019-ncov-eueea Northern ireland: https://www.health-ni.gov.uk/covid-19-statistics
So, in this particular case, it seems unlikely that population density is driving the differences.
I do agree that raw country comparisons can often mislead, but i stand by my original points.