Statistics updates is delayed when using `commit and chain`

Hello,

With a customer, we found out that when using `commit and chain`,
statistics on the table were not updated. Here are the steps to reproduce
(My customer saw this on Postgres 13, I confirmed it under current main
version):

laetitia=# laetitia=# select n_tup_ins from pg_stat_all_tables where
relname = 'test';

n_tup_ins

-----------

17

(1 row)

laetitia=# begin;

BEGIN

laetitia=*# insert into test (value) values ('bla');

INSERT 0 1

laetitia=*# commit and chain;

COMMIT

laetitia=*# select n_tup_ins from pg_stat_all_tables where relname = 'test';

n_tup_ins

-----------

17

(1 row)

laetitia=*# commit;

COMMIT

laetitia=# select n_tup_ins from pg_stat_all_tables where relname = 'test';

n_tup_ins

-----------

18

(1 row)

Before issuing the last `commit`, I used another connection to check the
value of the statistics from another transaction and it was not updated:

laetitia=# select n_tup_ins from pg_stat_all_tables where relname = 'test';

n_tup_ins

-----------

17

(1 row)

Maybe it's not a bug and it's on purpose but I can't understand what would
prevent the statistics collector from being triggered after a `commti and
chain`.

Have a nice day,

Lætitia