Attributive だろう

Question

I came across this sentence (下町ロケット, p.239):

「どうだ、そろそろウチに戻ってこないか」
そんなことをいいはじめたのは、六十五歳を過ぎ、体力的な衰えを感じはじめただろう頃だ。

Meaning-wise, it seems hard to imagine it means anything other than 感じはじめた頃だろう. However, I wonder, is there is a slight difference in nuance, or something else that would have made the author choose this rare (at least I don't think I've seen it before) usage of だろう?

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和訳

この文章に出くわしました：（下町ロケットp.239）

「どうだ、そろそろウチに戻ってこないか」
そんなことをいいはじめたのは、六十五歳を過ぎ、体力的な衰えを感じはじめただろう頃だ。

意味的には、「感じはじめた頃だろう」と同じだと思っていますが、これほど珍しい（少なくとも私は見たことのない）「だろう」の使い方をする理由（ニュアンスの違いなど）はありますか。

Answer 1

I think there is a slight difference in what the uncertainty is about:

1. … 六十五歳を過ぎ、<unsure>体力的な衰えを感じはじめた</unsure>だろう頃だ。
2. … <unsure>六十五歳を過ぎ、体力的な衰えを感じはじめた頃</unsure>だろう。

In #1, the uncertainty is less about the actual time frame, and more about what their physical condition had been. In #2, it seems that the uncertainty is mainly about the time frame.

Likewise, in the sentences below, #1 is unsure whether the children were hungry, while #2 is unsure whether these children had it worst:

1. いちばん大変だったのは、お腹が空いていたであろう子供たちだ。
2. いちばん大変だったのは、お腹が空いていた子供たちであろう。

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I did not know that this use of だろう is technically incorrect and should be であろう instead (as brought up by xeta217 in the comments). I could also find one page online that agrees with this rule. However, there are probably many other books that use phrases like だろう頃に, so it may be that this rule is not widely known, or debatable.