If focused on that, because the consonant shift is generally what shows us which words come from German vs. Germanic, rather than address the subsequent shift from g.
The softening of the "g" into "y", "j","i" is also a feature in other Germanic languages for many other words. E.g. in modern Norwegian "deg" (the object form of "you" / "thee" which is necessary in Norwegian still) can be pronounced both with a very pronounced g, and with a tiny shift of the tongue to soften it into a i/j. In English we have "dough" where the "g" has disappeared, but was still there in Old English, and where in modern Norwegian you again can choose /dɛɪːɡ/ or /dæɪ̯/. They all, again, share a root in Germanic with the German, where we again have a strong indication the other Germanic languages evolved their current versions from Germanic rather than German in that the German got a "t" from the consonant shift: "Teig" (vs Low German "Deeg")
Jepp, that's precis som på svenska, där “dig” uttalas “dej”. But AIUI you can see from old songs and poems, by which words it was used to rhyme with, that the spelling is how it used to be pronounced not so many centuries ago. (And yes, “dough" heter “deg” på svenska också. Så og på norsk, antar jag?)
The softening of the "g" into "y", "j","i" is also a feature in other Germanic languages for many other words. E.g. in modern Norwegian "deg" (the object form of "you" / "thee" which is necessary in Norwegian still) can be pronounced both with a very pronounced g, and with a tiny shift of the tongue to soften it into a i/j. In English we have "dough" where the "g" has disappeared, but was still there in Old English, and where in modern Norwegian you again can choose /dɛɪːɡ/ or /dæɪ̯/. They all, again, share a root in Germanic with the German, where we again have a strong indication the other Germanic languages evolved their current versions from Germanic rather than German in that the German got a "t" from the consonant shift: "Teig" (vs Low German "Deeg")