Find max sum of daily total within date range
You will need a calendar table to distribute the quantity to each date in the interval:
CREATE TABLE calendar (dt date not null primary key);
INSERT INTO calendar (dt)
VALUES ('2021-01-14'),('2021-01-15'),('2021-01-16'),('2021-01-17')
     , ('2021-01-18'),('2021-01-19'),('2021-01-20'),('2021-01-21');

Now you can use a join for that:
SELECT MAX(qt)
FROM (
    SELECT dt, SUM(quantity) as qt
    FROM timetable t
    JOIN calendar c
        ON c.dt BETWEEN date(t.dt_start) AND date(t.dt_end)
    WHERE c.dt BETWEEN '2021-01-14' AND '2021-01-23'
    GROUP BY dt
) tt;

A calendar table is very useful for many tasks so I suggest you create one and populate it. Many DBMS including newer versions of MySQL support recursive CTE:s which can be used instead.
Getting the totals for each dt:
SELECT dt, SUM(quantity) as qt
FROM timetable t
JOIN calendar c
    ON c.dt BETWEEN t.dt_start AND t.dt_end
WHERE c.dt BETWEEN '2021-01-14' AND '2021-01-23'
GROUP BY dt;

Fiddle