Sounds like they're just trying to sell you something.
First, a broken spoke is sometimes just a broken spoke. Replace the spoke and keep riding. If you keep having problems you have a decision to make. Have the existing wheel rebuilt correctly or have a new one built.
If you want a new bike, then by all means buy one. Just don't do it because you think you have to.
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BTW, my personal preference is to have the kid on the bike with me.