Human factors are important -- remember the Butterfly Ballots of the 2000 elections!
https://en.wikipedia.org/wiki/2000_United_States_presidentia...
I think that 0 being "don't care" and negative numbers being negative percentages is easier to grasp and more balanced and granular than (3) being neutral. Even -3 to 3 would be better than 1 to 7.
Voting "A: -100%, B: -100%, C: -100%" weights each candidate the same as "A: 0%, B: 0%, C: 0%" the same as "A: 100%, B: 100%, C: 100%", but as you said, it's useful to get an overall count of the voters' absolute satisfaction with each candidate.
A popular positively winning candidate would have a more legitimate claim to a mandate, and the lesser of two evils candidates would be branded as such.
I agree with you here, maybe I did not articulate my previous comment correctly to reflect this.
A benefit of cardinal systems that you are actually pointing out is that not only can we determine winners, but we can also get a lot of information about voter preference. This is quite useful to politicians and researchers (understanding how represented people feel). But I will also note that a strictly negative ballot is likely to be rare as it is purely a protest ballot.