"INSERT INTO res_partner () VALUES ()" failure (autoincrement in postgres)
Your linked question is for MySQL which doesn't really have a track record for valid standard SQL - and having an empty column list or empty values list is far from being "legal" syntax in standard SQL or Postgres.

If you want to insert the default values you have two choices

insert into public.res_partner (id) values (default);


Or if you do not want to specify the column name, use:

insert into public.res_partner
default values;