I have no idea how you got the [] round what appears to be a textual representation of a LINESTRING, and there is a fairly strong clue in the error message. To remove them:

 1. use [regexp_replace](https://www.postgresql.org/docs/current/functions-string.html) to remove the superfluous [ and ] around your text string.
 2. Use the Postgres [concat](https://www.postgresql.org/docs/current/functions-string.html) operator to add a leading **LINESTRING**, so that you have a valid textual representation of a geometry.
 3. Use [ST_GeomFromText](https://postgis.net/docs/ST_GeomFromText.html) to convert to a geometry. You can either wrap this with **ST_SetSRID**, as you have, or use the second form, adding in the SRID to the text string -- it makes no practical difference.
 4. Combine the **ALTER TABLE** statement with **USING**, as you have. This basically says change the data type in place, using, the updated geometry type in place of the existing text type.

For example, using a simplified example of what you have above, [(1 1, 2 2, 3 3)], as I am too lazy to type out yours:

    SELECT ST_GeomFromText(concat('LINESTRING',
                regexp_replace('[(1 1, 2 2, 3 3)]', '(\[|\])','', 'g')));

returns a linestring -- if you wrap this in ST_AsText, you get,

>  LINESTRING(1 1,2 2,3 3)

Note, the (a|b) syntax means match a and/or b the \ before the [ and ] is an escape character, as [ and ] have special a meaning in regular expressions, and g is a global flag, meaning replace everywhere.

Putting this altogether:

    ALTER TABLE area_grid_01 ALTER COLUMN coordinates TYPE Geometry(LINESTRING, 4326) 
    USING ST_SetSRID(ST_GeomFromText(concat('LINESTRING', 
               regexp_replace(coordinates, '(\[|\])','', 'g'))), 4326);


ought to do what you want, using your coordinates column instead of hard-coded numbers as above. The last bit is untested, so there may be an extra/missing ( or ) somewhere.