I would argue that it is not part of the schema information for two rather obvious reasons: first, it does not affect performance; second, in order to offer record counts, Athena, Presto, and Trino must process all data files and sources.
AFAIK The only alternative is to construct the query using SQL or another language, and then execute it because Presto/Trino does not support any type of procedural query execution (such PL/SQL paired with something allowing to run SQL from string). Let's start with this:
with tables(full_name) as(
SELECT '"' || t.table_schema || '"."' || t.table_name || '"' as full_name
FROM "information_schema"."tables" t
)
select array_join(array_agg('select ''' || full_name || ''' as table_name, count(*) as rows_count from ' || full_name), ' union all ')
from tables
group by true;
As an alternative, you can write a custom Athena function using lambda, which will execute the necessary SQL statement on the fly.