英文文法 不定詞與動名詞?

Even stubborn person wants to see every child getting quality education.

"getting" is the Object Complement of the noun phrase "every child", which is the object of the verb "see".

As you may have learned it before, the "sense verbs" such as see, hear, smell, etc. can take two kinds of object complement:
1. participle:
e.g. I saw a boy running across the road.
2. bare infinitive (without "to"):
e.g. I saw a boy run across the road.

getting 可以換成 to get 嗎?
Yes, but you have to omit the particle "to".

Even stubborn person wants to see every child get quality education.

(這裡的getting應該也不是who gets 的縮減) ~ correct!