How to insert random data into a SQLite table using only queries?

You can use a recursive query.

This query generates a table with 1000 random numbers:

CREATE TABLE test(field1);

INSERT INTO test
  WITH RECURSIVE
    cte(x) AS (
       SELECT random()
       UNION ALL
       SELECT random()
         FROM cte
        LIMIT 1000
  )
SELECT x FROM cte;

Going for hacky, this'll insert up to 2048 random rows in one go, decide how many by setting the limit;

INSERT INTO TEST (x,y,z)
  SELECT RANDOM(), RANDOM(), RANDOM()
   FROM (SELECT * FROM (
         (SELECT 0 UNION ALL SELECT 1) t2, 
         (SELECT 0 UNION ALL SELECT 1) t4,
         (SELECT 0 UNION ALL SELECT 1) t8,
         (SELECT 0 UNION ALL SELECT 1) t16,
         (SELECT 0 UNION ALL SELECT 1) t32,
         (SELECT 0 UNION ALL SELECT 1) t64,
         (SELECT 0 UNION ALL SELECT 1) t128,
         (SELECT 0 UNION ALL SELECT 1) t256,
         (SELECT 0 UNION ALL SELECT 1) t512,
         (SELECT 0 UNION ALL SELECT 1) t1024,
         (SELECT 0 UNION ALL SELECT 1) t2048
         )
    ) LIMIT 246;