That is not a silly question at all.
Here are 3 explanations to help you understand better the use of “ng”:
1. Basically, “ng” functions as the preposition “of” in a sentence. It is used to express a relationship or ownership.
That function appears in two of your given sentences – “binti NG tatay (leg OF the father)” and “pangulo NG bansa (president OF the country)”.
2. Tagalog verbs are unilaterally just verbs to native Tagalog speakers. However, when we teach Tagalog to English speakers, the concept of subject/actor- and object-focus verbs enter the picture.
“Niyay
akap ng bata ang bint
î ng tatay niya” uses an Object-focus verb (“niyay
akap”, inf. is “yak
apin”). If we’d use the Actor-focus verb in that sentence, it will become “Yumay
akap ang bata sa binti ng tatay niya”. The infinitive of “yumay
akap” is ”yum
akap”.
Both sentences mean the same thing – The child is hugging/clinging to the leg of his father.
The Actor (bata) and the Direct Object (binti) are the same in both sentences. The verbs are different though.
Restating the sentences:
“Yumayakap ANG bata SA binti ng tatay niya” becomes -
“ANG bata ay yumayakap SA binti ng tatay niya (The child is hugging/clinging to the leg of his father)”. Notice that here we have the actor/subject-verb-direct object order of the elements of the sentence. This is the same as the form of the English active voice.
“Niyayakap NG bata ANG binti ng tatay niya” becomes -
“ANG binti ng tatay niya ay niyayakap NG bata (The leg of his father is being hugged/clung to by the child)”. The elements of this sentence now follow the object/subject-verb-actor order. This is like the form of the English passive voice.
(Take note of the "ang" in the two sentences. The words that follow them are the subjects of the sentence.)
When the Actor is the subject of the sentence, the verb used is Actor-focus; when the Object is the subject of the sentence, the verb used is Object-focus.
AND SO, when a sentence uses an Object-focus verb, the “ng” is used to mark the Actor of the verb. The “ng” relates or gives “ownership of” the action to the Actor. You may think of “ng” in such cases as the preposition “by” – being hugged/clung to BY the child.
3. The “na” is used as a linker for a modifier. The rule is that when the word before the “na” ends in a vowel, you may attach the “na” to the preceding word as “-ng”. (Although we apply the rule almost all the time, it is not compulsory and will not create an error if not followed).
bago NA pangulo = bagoNG pangulo
pangulo NA bago = panguloNG bago
gusto NA magalit = gustoNG magalit (wanting to get angry)
magsisilbi NA pangulo = magsisilbiNG pangulo (will be serving as president)
In “gustoNG magalit” and “magsisilbiNG pangulo” they may actually be connected to the words they are modifying had they not been separated by other words between them. That though does not change the actual purpose of the modifier. If we’d move out the intervening words or remove the adjunct words from the sentences, we’d have:
GUSTO mo ba akoNG magalit? => Ikaw ba GUSTO NA MAGALIT/GUSTONG MAGALIT ako? - “Mo” needs to be changed to “ikaw” to move it out of the required word order when “mo” is used.
MAGSISILBI siya bilang bagong pangulo ng bansa. (He will serve as the new president of the country.) => MAGSISILBI NA/MAGSISILBING pangulo siya ng bansa. (He will serve as president of the country.) - The "na (as)" does not appear in the longer sentence because "bilang", which also means "as" took its place.
Therefore, this “ng” is not really “ng” but a transformed “na”.
by: 
