The fact you have the private key permits access, not possession of the public key even though the public key is the one you add to the file on the server side (asymmetric cryptography). The "secure way" may also suggest that you have to keep your public key secret which is not true.
I believe the statement is most charitably interpreted as the channel must be secure against modifications so that when you are attempting to put the public key in the file that some other contents aren't inserted instead.
On the other hand, SSH gives you just that if you verify the fingerprint which is not mentioned in the guide. MITM-proof channel is needed if one gains any kind of shell access because then public key substitution can happen in various ways, even without the user him/herself explicitly editing the file. Furthermore, if one allows MITM when accessing shell, substituting the public key is just one of very serious security problems.
Certainly many other operations require a tamper-proof channel, and many more things can go wrong without one but I believe the author was trying to indicate the specific requirement here rather than making a specifically incorrect statement.
