Using a free public SMS number for LinkedIn is basically begging to get banned. I've seen it happen more times than I can count.
Their security systems aren't stupid. They flag VoIP numbers and recycled public pool numbers instantly - those numbers have been burned by spammers and bots a thousand times over. The real trap is the recovery check months later. LinkedIn loves to spring random verification prompts. if that public number's gone - and it will be - you've permanently lost the account.
Want privacy without losing access? skip the free garbage. Get a paid, non-VoIP burner number from something like Cloaked or Tossable Digits. A clean, dedicated line you control. keeps your personal phone private, but you keep access whenever LinkedIn forces another security check. That's the only sane play.