This is actually an interesting question because I have also heard these explanations and yet sometimes ~出そう seems natural..
I found
a paper called 開始の「~出す」「~始める」について : 小説作品における用例分析を通して前接する動詞の傾向を探る - basically looking at how these forms are used in novels and which verbs combine with these suffixes.
Section 5 is called 「〜出す」と意志表現. Basically, the author starts by discussing the fact that it is often stated that one of the differences between 出す and 始める is that 出す isn't used for volitional - or at least not commonly used (some quotes brought up say things like 現れにくい or あまり使われません).
However, some counter examples are brought up. The first quote actually uses 歩く:
ぼくはくるりと後ろを向いて、歩き出そうとした。
The author of this paper splits the cases they found into three broad types:
1) 〜出そう (expressing volition)
Found in the texts they analysed with these verbs: 駆ける・歩く・走る・笑う
All cases in their corpus used "出そうとした"
2) 〜出したい (expressing desire)
Found with 泣く、笑う (e.g. 泣き出したくなる)
3) 〜出すつもり (expressing intent)
Found only with 咲く and explained as a special case.
They also looked for examples with 言う and 動く as these were verbs often used with 出す (but where volitional type sentences weren't found in their initial text analysis), and found examples of those as well.
The conclusion they give is simply that while it most cases ~出す isn't used with volitional, for certain verbs, it is not unnatural to do so. The cases where ~出す + volitional is used are almost all cases of verbs where the combination of that verb with 出す is common.
My guess: in these cases the meaning of the combination has drifted from what you might expect by just combining the verb with 出す (in particular, 言い出す has it's own dictionary entry), and therefore they don't necessarily adhere to the "rules" of ~出す (they're more like guidelines?).