綻び describes something coming undone, such as fabric fraying or hidden flaws starting to appear. In the song, 「次第に綻びボロが出てきたとして」 means “even if things gradually unravel and our flaws begin to show,” giving this poetic word an intimate, human meaning.
Small Planet starts with the most relatable complaint in pop music: ずっと夢を見ていたいのに [I want to keep dreaming forever], but 決まった時間に今日も / 叩き起こされるんだ [at the same set time, today again, I get dragged out of bed]. The singer just asks for one thing, 大きな欠伸くらい許してよ [at least let me have a big yawn]. Then comes a lovely joke: short of oxygen, 産声をあげているんだ [I'm letting out a newborn's first cry]. Every grumpy morning is a tiny birth.
The "small planet" is not outer space. It is いつもの街角 [the usual street corner] and a patch of sunlight nobody notices, where 凍った愛が静かに溶け出したんだ [frozen love quietly started to melt]. Spring sneaks in as a smell from the town, and the chorus admits 結局何処へも行けやしない僕らは [in the end we can't go anywhere], then sneezes at the winter wind. That sneeze is the best detail in the song. By the end, even if things start to fray and ボロが出てきた [our flaws start showing], the answer is a shrug, まあそれも別にいいかと [well, that's fine too], and two people くしゃみをして笑い合うのさ [sneeze and laugh together]. Worth learning: 産声 (ubugoe) is specifically a baby's first cry at birth, and ボロが出る is an idiom for your hidden faults coming out.