I have noticed that the default list of header to be signed are as below.
sender, from, to, cc, date, subject, mime-version, content-type, message-id, references, in-reply-to

reply-to is missing from the list. Is there a reason this is missing, or just an oversight?

reply-to is included in the recommended list within the DKIM RFC 

Is there any reason I should not add this header into the signed list manually?

Brian

No problem, you can add Reply-To in the header list, there is no negative effect.
But it is not a MUST, sometimes the transport agent will change the reply-to header based on the user rule, that is why we don't use it by default.