I am wondering if words using the suffix "ing" should be pronounced with a silent "g"?
The short answer is "Yes, in -ing the <g> is silent." But that does not mean that it rhymes with in or spin. Although in informal speech we often pronounce -ing as if it were spelled <in'>, as in "Singin' in the rain," the full pronunciation of -ing is not [in]. To explain why, we have to look a bit at the history of the English language.
In Old English the letter <n> spelled two different sounds. The first and most common sound was the [n] that it usually spells today, as in noon. This [n] is pronounced well forward in the mouth: We place our tongue against our upper dental ridge and release the sound through the nose. But in front of the two sounds [g] and [k], two sounds pronounced well back in the mouth, with the tongue up against the velum, the <n> spells a different sound. In Old English the second sound spelled <n> was also pronounced well back in the mouth, in anticipation of upcoming [g] or [k]. This produced -- and today still produces -- a nasal sound different from [n], a sound called eng (suggesting its sound) and represented in most dictionaries as an <n> with a long, curved right leg: . This is the sound we hear today at the end of words like sing and thing.
Old English words like ðencan "to think" and singan "to sing" would contain the string of sounds [k] or [g], as in a few modern words like wrinkle and single .But over the centuries during the evolution from Old English through Middle English to Modern English, in most words the [g] fell silent -- including in the suffix -ing. So nowadays the full pronunciation of -ing is [i], with no separate [g].
The larger spelling principle involved here is that today eng is spelled <ng> everywhere -- except in front of [k] or [g]: sing but sink and single.