> You could simply let anyone vote for anyone and elect the top N contenders where there is N seats
You could, but that would actually be worse the FPTP in single member districts.
> Logically the choices would resemble the choices of the population.
No, they wouldn't, except by chance. Say you have six candidates, A-F, for three seats. 75% of the electorate prefers an ideology generally espoused by A, B, and C; 25% prefers an ideology generally espoused by D and E, and 10% prefers an ideology espoused by F.
In the first group, A is by far the most effective campaigner and gets 70% of the total vote, with the remainder split evenly between B and C; in the second group, support is split equally between D and E. The last, of course, all goes to F.
So, in your top 3 vote, A and F win seats in the clear, and the third vote goes to a tiebreaker between D and E; this does not at all reflect the distribution of views in the population.
That's why methods like Single Transferrable Vote and it's relatives, using preference rather than bullet ballots, exist.