How to split OSM roads into individual segments at intersections?

Simple answer: Don't. You shouldn't do it that way.

From the OSM road Shapefiles, it is impossible to distinguish between intersections and over/underpasses. You'll create intersections that don't exist in reality if you split all seemingly crossing roads.

You'll need to get your hands dirty with the original OSM file, if you don't want to use existing tools such as osm2pgrouting (where the network is small enough) or osm2po.

Not a real solution to your problem, but try osm2po ... it creates perfect SQL code for routing in pgrouting: http://osm2po.de/

About your general problem, using pgRouting: I think @Uffer, @GisStudent and others that are showing how to use "OSC & etc.", they are right. Go following the clues of of "best practices" and "standards"...

About your question: "split roads into individual segments at intersections" or "how to remove all of the original unsplit geometries". I can help if you show here your results here, step-by-step...

First step: topology analysis

 CREATE TABLE split_points_topo as
  SELECT     
    a.gid as gid_a, b.gid  as gid_b, ST_Relation(a.geom, b.geom) as DE9IM_code
  FROM
    roads as a,
    roads as b
  WHERE a.gid != b.gid AND a.geom && b.geom;
 
 SELECT DISTINCT st_geometryType(geom) FROM roads;
 SELECT DISTINCT DE9IM_code FROM split_points_topo;
 -- list here the results o these two queries!  ... after we can continue.