I voted yes.
Reason 1. At the moment any non-scoring constructor at the end of the year has their final position and therefor their money decided by their highest single race finishing place in the season. So if 2 constructors score no points all year, 1 lucky race could see the better of the 2 teams finish last, with the lucky team finishing ahead. With the introduction of more teams in the future this injustice could become more frequent. If everyone scored points then it would be very clear who is the superior team and what order they are in the championship.
Reason 2. It would make team mate comparisons very easy to follow particularly for teams further down the grid as their drivers regularly miss out on points. All without having to look out best finish statistics for both drivers from F1 nerd sites.
Reason 3. It gives the drivers at the back of the field extra incentive to keep on racing. If you have 15th place but have already had an 11th place finish for your team this year there is zero point taking on a risky overtake on at the last lap. But if every position gets points, then ever place counts, no matter what you have already achieved in previous races.
Reason 4. It makes the drivers championship more accurate. In an extreme example, lets say its the last race. The championship is tied between LH and SV. Hamilton and Vettel have a first lap incident at a track where you can't overtake easily. Hamilton only recovers to 15th Vettel recovers to 11th. Under the current system their fight back is utterly irrelevant as nether of them score points. Yet such a fight back would be a championship decider if points were awarded to everyone. Under the current system the championship would go the the person with the most pole positions. Not counting results below 10th is just stupid in this type of scenario.