count how many event between dates sql in different rows

It's not entirely clear what your "use only one table" restriction means.

Let's say you somehow got this table date_series:

Using this table, count by self JOIN date_series.
We take DISTINCT time_series for case where several reservations have same date.

select  t1.time_series,count(t2.start_date) as No_of_rooms
from (select distinct time_series from date_series) as t1
left join date_series t2 on t1.time_series between t2.start_date and t2.end_date
group by t1.time_series

fiddle

A more common way to solve such a problem

1. Generate date_range. In this example by recursive query. More examples see reference in @JNevill comment.
2. Join source table with date_range, then count(start_date) thru GROUP BY.

See exmple

with recursive date_range as(
  select cast('2024.01.14' as date) as report_date, cast('2024.01.19' as date) to_date
  union all
  select date_add(report_date,interval 1 day), to_date
  from date_range
  where report_date<to_date
)
select report_date,count(start_date) as No_of_rooms
from date_range d
left join reservations r on d.report_date between r.start_date and r.end_date
group by report_date

fiddle

Your intermediate table somewhat is the dump of what you would have got from a CTE.

Joining the table to itself, by considering only the first column in its first (left) use, and only the last two columns on the right side, will emulate the two original tables.

select t.time_series, count(1) no_of_rooms
from ts t join ts f on t.time_series between f.start and f.end
group by t.time_series
order by t.time_series;

See it in a fiddle.

Note that the comparison to the null-valued columns of the right side will naturally discard rows for that second table when the row has only a first table use (e.g. the 16.01 which is not a booking occurrence but only a series date).

Handling of same-date bookings

Note that this solution entirely relies on the careful building of your intermediate table.
In particular when two bookings start as the same date only one should get a valued time_series, the other one shall have a null in that column.

See a fiddle with another booking to illustrate that.

IMHO

That "only one table" requirement looks suspicious, because it complicates everything and essentially forces to recompose (deduce) the original table + the series from your intermediate table.

On the contrary, the requirement may be a way to force you to not create another, intermediate table, to direct you to a full-CTE solution (including the series generation):
instead of create table series_and_bookings as select …; select [final query]; it would like to incite you to write with series_and_bookings as (select …) select [final query];.
Really we can say that it doesn't add a table (in case of a crash a CTE won't let artifacts, which I think is the real requirement).

Thus you may want to challenge the requirement, at least ask for a clarification.