Why is this PostgreSQL not null constraint not catching this insert?
As the check constraint with the regex will only validate non-null values, I would split this up in three different check constraints:

create table application_user(
    -- other columns, primary key, etc.
    phone text,
    email text,
    constraint exactly_one check (num_nonnulls(phone, email)=1 ),
    constraint valid_email check (email ~ '(?:[a-z0-9!#$%&amp;''*+/=?^_`{|}~-]+(?:\.[a-z0-9!#$%&amp;''*+/=?^_`{|}~-]+)*|"(?:[\x01-\x08\x0b\x0c\x0e-\x1f\x21\x23-\x5b\x5d-\x7f]|\\[\x01-\x09\x0b\x0c\x0e-\x7f])*")@(?:(?:[a-z0-9](?:[a-z0-9-]*[a-z0-9])?\.)+[a-z0-9](?:[a-z0-9-]*[a-z0-9])?|\[(?:(?:25[0-5]|2[0-4][0-9]|[01]?[0-9][0-9]?)\.){3}(?:25[0-5]|2[0-4][0-9]|[01]?[0-9][0-9]?|[a-z0-9-]*[a-z0-9]:(?:[\x01-\x08\x0b\x0c\x0e-\x1f\x21-\x5a\x53-\x7f]|\\[\x01-\x09\x0b\x0c\x0e-\x7f])+)\])'::text),
    constraint valid_phone check (phone ~ '\d{10}'::text)
);


check (num_nonnulls(phone, email)=1) only validates that either one of the column is specified. The other two only check the format of the respective column. This - I think - makes things a bit easier to understand.