If the intention is "increase believability a real human wrote this bot message" they do

Sure but adding error rate also makes it so you trust the speaker less, because they aren't educated enough or ESL.

Since sprinkling some mistakes makes you appear human, as opposed to a spam bot, people would be trusting you more, not less.

Not to mention the relatability aspect ("he's a plain dude, making mistakes and all, not some over-educated schmuck")