Rnm4, Glad to meet someone who is also interested in Frege.
He wrote this article 'Uber die Algemeinheit' (generality)
to explain the use of expressions like 'all', 'some', etc.
and never used expression 'quantifier' because this expression was not used in his time. You can't attribute your first statement to Frege as far as I know.
Regarding the 'context' : 'Nur in Zusammenhang des Satzes
hat ein Wort Bedeutung' is the so called 'compositionality principle' . Ie a sentence is an composition , composed from different parts so to anylise a sentence one need to 'decompose' it. Also an sentence need to be 'abgeschlossen' . Ie a sentence need to be a complete whole. Frege was primary interested in 'scientific sentences' those which are true or false. He wanted to construct an scientific language. Now consider an mathematical or particle physics statement. Should the mathematician also state that his sentence is in the 'context' of mathematics while our physicist should add that his statement is in the context of physics?
Why should Frege provide any explanation of sentences which
are not truth functional? Say 'propositional attitudes'.
He wrote about 'common language' only to illustrate why
'we' need according to him a more precise language.
His Begriffsschrift (aka new logic) was meant as such.
Regarding Frege-Wittgenetein you should read their correspondence in which Frege made comment about Tractatus.
There you can also find Frege's remarks about Wittgensteins use of 'S is P' sentence form.
Not to avoid your statement: ''no ,you can't analyse every
sentence because you have read Frege'' I must admit that you are right.
Regards,
