Don't teach your child pointless kid's songs that have been sung to you. Instead, compose a few that are about important life lessons you want your child to remember for life.
In my area, people teach kids the same folk songs that have been passed down for generations. Many have completely pointless, absurd, outdated, or even dark messages that have been influenced by plague, war, and other calamities. They are sung to kids repeatedly until they are memorized for life. My mind is full of brain-spam from childhood.
When singing to your kids, don't just repeat the same songs that live rent-free in your mind. Consider the message each song brings.
Instead, compose a list of important lessons you would like your kids to remember for life. If you can, come up with a children's song for each message. If you can't - find someone who is good at it and let them put together a few.
Maybe one day these songs will influence your child's decisions in a more favorable direction.