This is possible using a different 'geometry generator' to the one that you originally intended I suspect, You can avoid having to generate the all connections line layer by using a virtual layer:

Some points:

[![sample points][1]][1]

Add a virtual layer using the following SQL - the JOIN matches every point to every other, and the line geometry is generated using the Spatialite [MakeLine][2] function:

<!-- language: lang-sql -->

    SELECT s.id 'ID1', c.id 'ID2' , MakeLine(s.geometry,c.geometry) 'geometry'
      FROM SamplePoints AS s JOIN SamplePoints AS c 
      WHERE s.id <> c.id

The result:

[![Points with connecting lines][3]][3]

This virtual table could then be filtered dynamically using the Atlas functionality.

Dynamically generating curved lines would be trickier, but should still be possible. Curving them in a cartographically appropriate way (considering the distribution of the other lines) is probably beyond what can be achieved with virtual layers.

Update:

With a little tinkering and a lot of reference to the Spatialite function list I have generated curved lines:

[![Points with curved connecting lines][4]][4]

The virtual layer SQL for this is below. Note that all permutations are shown, and reversing the order of the start and end location generates the complementary curve.

<!-- language: lang-sql -->

    SELECT s.id 'ID1', c.id 'ID2' , 
    MakeArc(
    	X(Project(MakePoint((X(s.geometry)+X(c.geometry))/2,(Y(s.geometry)+Y(c.geometry))/2),distance(s.geometry,c.geometry),Azimuth(s.geometry,c.geometry) - PI()/2)),
    	Y(Project(MakePoint((X(s.geometry)+X(c.geometry))/2,(Y(s.geometry)+Y(c.geometry))/2),distance(s.geometry,c.geometry),Azimuth(s.geometry,c.geometry) - PI()/2)),
    	distance(s.geometry,c.geometry) * 1.1180339887,
    	90 - azimuth(Project(MakePoint((X(s.geometry)+X(c.geometry))/2,(Y(s.geometry)+Y(c.geometry))/2),distance(s.geometry,c.geometry),Azimuth(s.geometry,c.geometry) - PI()/2),s.geometry) * (180/PI()),
    	90 - azimuth(Project(MakePoint((X(s.geometry)+X(c.geometry))/2,(Y(s.geometry)+Y(c.geometry))/2),distance(s.geometry,c.geometry),Azimuth(s.geometry,c.geometry) - PI()/2),c.geometry) * (180/PI())) 'geom' /*line:27700*/
      FROM SamplePoints AS s JOIN SamplePoints AS c 
      WHERE s.id <> c.id


  [1]: https://i.sstatic.net/JRQX1.png
  [2]: http://www.gaia-gis.it/gaia-sins/spatialite-sql-4.2.0.html#p0
  [3]: https://i.sstatic.net/gFQKv.png
  [4]: https://i.sstatic.net/ntwIM.png