Sorry I must not have explained my question well.
There is a day side of the earth, and a night side. The moon should be on the day side for 14.75 days and on the night side for 14.75 days since it orbits the earth every 29.5 days. Yet the moon can be observed at night for more than 14.75 nights in a month. The entire lunar phase can be observed in a month as long as the sky's clear.
No, we understood quite fine.
By "The moon is on the day side for 14.5 days", what that really means is the sub-lunar point should be on the day side.
Your argument still relies upon only being able to see a tiny sliver of the sky.
You may as well be saying we should only be able to see the sun at solar noon.
As a simple example, consider a quarter moon.
For this, the angular separation between the sun and moon is ~ 90 degrees.
This means the moon will reach its peak in the sky at sunrise or sunset (depending on if it is waxing or waning).
It wont magically vanish at sunset when it was high in the sky just before it.
It will still be visible after sunset.
Likewise, it wont magically appear high in the sky just after sunrise. Instead it will be visible before sunrise.
Here is a diagram to show this:

The moon is clearly above the day region of Earth, but it can still be seen from the night region.
There is no issue here.
If you think there is, perhaps you can draw a diagram to explain it better?
In fact, a full moon shouldn't be possible at night because the moon goes into Earth's shadow when it's on the night side.
The full moon is rarely exactly opposite the sun.
This is because the Moon's orbital plane is tilted with respect to Earth's orbital plane, roughly 5.145 degrees.
This is why eclipses only occur in some parts of the year, because that is when the moon lies in Earth's orbital plane (or close enough to it).
And this applies for both solar and lunar eclipses.
Here is another diagram, to show what happens (very much not to scale):

The orange circle is the sun, the blue circle is Earth and the grey circle is the moon.
The purple line represents the edge of Earth's shadow (specifically the penumbra).
If the moon is above this line, it will not be in Earth's shadow.
I will be using these numbers:
Re: 6371 km (radius of Earth)
Rs: 695700 km (radius of sun)
Rm: 1737 km (radius of moon)
Des: 150000000 km (distance between Earth and sun)
Dem: 362600 km (distance between Earth and moon, chosen at perigee, for most difficulty keeping out of the shadow)
The angle a is found from:
sin(a)=(Rs+Re)/Des
This gives us a=0.27 degrees.
The angle c is obtained from:
sin(c) = (Re+Rm)/Dem
This gives us c=1.28 degrees.
The angle b is simply a+c.
This gives us 1.55 degrees.
i.e. if the alignment is such that the moon is 1.55 degrees out of the orbital plane during a new moon, it wont produce a lunar eclipse.
If I use the moon at apogee, it drops to 1.41.
And as said above, the orbital plane of the moon is offset by 5.145 degrees, much larger than the needed 1.55 degrees.
So it will not go into Earth's shadow for most of the year.
As a comparison, with a stated orbital period of 29.5 days, that means each day the moon will move by 12.2 degrees. Or alternatively, 1.55 degrees would correspond to roughly 3 hours. So doubling that (for both sides) corresponds to 6 hours, or roughly the longest duration of a lunar eclipse, such as what this one was:
https://en.wikipedia.org/wiki/July_2000_lunar_eclipse(the duration of the penumbral eclipse, from P1 to P4, not the duration of the total eclipse, nor the partial umbral eclipse)
And as you may be able to guess from how much the moon moves in a day and how people don't notice it visibly change, that 5.145 degree of tilt won't produce a noticeable effect on how full the moon appears.
Even using the full 12.2 degrees, viewing the moon as a circle, and considering a line going straight across the surface, only ~2% will be dark.
The only time the moon isn't visible at night is when it's in the new moon phase. It lasts for 2 nights. This is supposedly when it's in Earth's shadow.
That is NOT when the moon is in Earth's shadow.
It is in Earth's shadow during a lunar eclipse, which occurs during a full moon.
During a new moon, it is as close to the sun as possible, and will sometimes cause a solar eclipse.
The moon's phases are not caused by Earth's shadow, but by the direction to the sun illuminating a particular region of the moon.
With the angular separation between the sun and Earth (at the moon) dictating the phase, with that angle being related to the angular separation between the sun and moon on Earth, by the 2 angles approximately adding to 180 degrees.
When the moon and sun are separated by ~0 degrees, it will be a new moon.
When the moon and sun are separated by ~180 degrees, it will be a full moon.
When the moon and sun are separated by ~90 degrees, it will be a quarter moon.