Found that out when my dad died in 2000, and we asked for one - we didn't expect foul play but he'd struggled with alcohol problems in the past and for closure we wanted to understand if he'd gotten back into drinking and that might have been why he died (he had heart problems and were on drugs that would have made combining them with alcohol a likely cause).
The coroners office made it very clear that that since it was clear he died from his heart stopping and there was no immediate reasons to suspect that there were any external factors that'd make a difference to the primary reason of death and no suspicion of foul play, an autopsy was out of the question unless we were prepared to cover the full costs (which were fairly steep).
That he may or may not have taken his heart medicine or may or may not have combined it with alcohol was to them irrelevant as long as it didn't change the main cause and they had no reason to see the distinction as relevant.
If there was doubt about the major cause, such as e.g. if he'd had a head wound or something else that might have indicated another cause, it might have been different.
This might very well also vary from coroners office to coroners office in some countries depending on capacity. This was in Oslo, so would have been one of the busiest areas.
Really? That's interesting. In neighbouring Finland, an autopsy is standard practice whenever the medical cause(s) of death is not completely clear.
Anyway, it's a relevant point that you make about death being caused by multiple factors, and crediting the causes in statistics is a game that can be played.