Short answer: Reproject them to World Sinusoidal (EPSG:54008), and use
-ts 7000 7000.
Disclaimer: By the statement of the question I am going to consider that your points are distributed in quadrangular form on the terrestrial globe. The same, I consider that a point every 7km covering the earth suface, is a huge amount of points. But this is how the question is written and I am going to stick to only the discussion on how to maintain the spatial relationship.
You know that if two runners, distanced 7km from each other and located on the equator, begin to run North, their trajectories cross. The distance between them will not be more or less 7km during the journey, it will decrease to 0. Therefore, the first thing that we must set aside if we want equidistant and orthogonal trajectories with each other, on a curved surface, is the idea of the meridians.
Meridians are useful because they all point to the north. But if we want to obtain a quadrangular distribution of points on the earth's surface, we must work with parallels in both directions. Parallel to the equator, and parallel to a central meridian.
I present an orthographic projection of points distributed quadrangularly on the Earth's surface, at a distance of 1500km from each other.
And that's how they look in the Geographic Coordinate System WGS 84:
But a raster file needs a quadrangular equidistant reference to represent the values of the pixels. Therefore it is necessary to project the points with a Sinusoidal (Sanson-Flamsteed) projection.
The trajectories parallel to the central meridian are seen as vertical lines in this system.
This is how the same points are seen, projected in the World Sinusoidal (EPSG: 54008) CRS.
Therefore the proposed solution is: Reproject them to World Sinusoidal (EPSG:54008), and use
-ts 7000 7000.