Apply function to all columns in a Postgres table dynamically

Question

Using Postgres 13.1, I want to apply a forward fill function to all columns of a table. The forward fill function is explained in my earlier question:

• How to do forward fill as a PL/PGSQL function

However, in that case the columns and table are specified. I want to take that code and apply it to an arbitrary table, ie. specify a table and the forward fill is applied to each of the columns.

Using this table as an example:

CREATE TABLE example(row_num int, id int, str text, val integer);
INSERT INTO example VALUES
  (1, 1, '1a', NULL)
, (2, 1, NULL,    1)
, (3, 2, '2a',    2)
, (4, 2, NULL, NULL)
, (5, 3, NULL, NULL)
, (6, 3, '3a',   31)
, (7, 3, NULL, NULL)
, (8, 3, NULL,   32)
, (9, 3, '3b', NULL)
, (10,3, NULL, NULL)
;

I start with the following working base for the function. I call it passing in some variable names. Note the first is a table name not a column name. The function takes the table name and creates an array of all the column names and then outputs the names.

create or replace function col_collect(tbl text, id text, row_num text)
    returns text[]
    language plpgsql as
$func$
declare
    tmp text[];
    col text;
begin
    select array (
            select column_name
            from information_schema."columns" c
            where table_name = tbl
            ) into tmp;
    foreach col in array tmp
    loop
        raise notice 'col: %', col;
    end loop;
    return tmp;
end
$func$;

I want to apply the "forward fill" function I got from my earlier question to each column of a table. UPDATE seems to be the correct approach. So this is the preceding function where I replace raise notice by an update using execute so I can pass in the table name:

create or replace function col_collect(tbl text, id text, row_num text)
    returns void
    language plpgsql as
$func$
declare
    tmp text[];
    col text;
begin
    select array (
            select column_name
            from information_schema."columns" c
            where table_name = tbl
            ) into tmp;
    foreach col in array tmp
    loop
        execute 'update '||tbl||' 
                set '||col||' = gapfill('||col||') OVER w AS '||col||' 
                where '||tbl||'.row_num = '||col||'.row_num
                window w as (PARTITION BY '||id||' ORDER BY '||row_num||') 
                returning *;';
    end loop;
end
$func$;

-- call the function
select col_collect('example','id','row_num')

The preceding errors out with a syntax error. I have tried many variations on this but they all fail. Helpful answers on SO were here and here. The aggregate function I'm trying to apply (as window function) is:

CREATE OR REPLACE FUNCTION gap_fill_internal(s anyelement, v anyelement)
  RETURNS anyelement
  LANGUAGE plpgsql AS
$func$
BEGIN
RETURN COALESCE(v, s);  -- that's all!
END
$func$;

CREATE AGGREGATE gap_fill(anyelement) ( 
  SFUNC = gap_fill_internal, 
  STYPE = anyelement 
);

My questions are:

1. is this a good approach and if so what am I doing wrong; or
2. is there a better way to do this?