I don't think the problem is in using the entire population as the denominator or just fatalities on regional roads as the numerator. That seems most consistent with the Swedish numbers to me.
That doesn't mean its an apples-to-apples comparison though because like you say the usage of the road network is going to vary and even the types of roads are going to be very different. Something like fatalities per mile driven seems more fair. And you could also probably break that down by urban, rural, highway, etc. roads.
In the end though, these comparisons are always going to be pretty flawed. Looking at the trends can be useful, but.... if you're just looking at any single high-level comparison its almost always going to be flawed.
That doesn't mean its an apples-to-apples comparison though because like you say the usage of the road network is going to vary and even the types of roads are going to be very different. Something like fatalities per mile driven seems more fair. And you could also probably break that down by urban, rural, highway, etc. roads.
In the end though, these comparisons are always going to be pretty flawed. Looking at the trends can be useful, but.... if you're just looking at any single high-level comparison its almost always going to be flawed.