Use whichever channel makes the outreach easiest to receive and respond to; there is no universal winner.

Email is often better for a specific, professional request when you can find the address, while LinkedIn is useful when email is unavailable or the connection is naturally social.

The quality of the message matters more than the platform: clear context, one reason for contacting them and a modest ask.

Do not send the same long template through both channels at the same time. For more detail, see the consulting cold email templates.

imo A short follow-up is fine. Repeated follow-ups started to feel awkward for me.

Email worked better when I had a specific reason for contacting that person.