Do all one-syllable words need to be followed by a 子 to make it colloquial?
Modern Chinese loves disyllabic words, that's why often you find seemingly meaningless "affixes" added to characters to sound "nicer".

Prefixation:


老 (老虎),
小 (小李),
初 (初一)


Suffixation:


子 (椅子),
頭 (木頭)


are just a few that are commonly used. Other ways of turning monosyllabic words into disyllabic words includes:


Compounding: putting together 2 roots with lexical meaning in forming a new word. E.g. 菊 (Chrysanthemum) -> 菊花 (Chrysanthemum flower)
Reduplication: duplicating a character or multiple characters. E.g. 爸 (Dad) -> 爸爸 (Dad)


So when you speak, try to observe how others usually collocate words. However, it is totally acceptable to use monosyllabic words when surrounded by enough sounds to make sense of it. So if they are just words used on their own, others may not totally understand them without context, since there are many homonyms.

In your example, 鋸 (beware the tool "saw" has a 金 radical) and 鏟 may match up with 子 to form 鋸子, 鏟子. These are good on their own. Or by making them compounds with description: 電鋸 (electric saw = chainsaw) and 鐵鏟 (metal spade) is also fine.

I believe the reason disyllabic words are more commonly used is simply because speaking in monosyllabic words may not convey the meaning well enough even with tones, so people tries to collocate characters together for making conversations easier. Despite all this, using monosyllabic words aren't "wrong". They are just rarely used and may make a sentence seem weird if used inappropriately.