If choosing between OAuth versions, go with OAuth 2.0.
OAuth bearer tokens should only be used with a secure transport.
OAuth bearer tokens are only as secure or insecure as the transport that encrypts the conversation. HTTPS takes care of protecting against replay attacks, so it isn't necessary for the bearer token to also guard against replay.
While it is true that if someone intercepts your bearer token they can impersonate you when calling the API, there are plenty of ways to mitigate that risk. If you give your tokens a long expiration period and expect your clients to store the tokens locally, you have a greater risk of tokens being intercepted and misused than if you give your tokens a short expiration, require clients to acquire new tokens for every session, and advise clients not to persist tokens.
If you need to secure payloads that pass through multiple participants, then you need something more than HTTPS/SSL, since HTTPS/SSL only encrypts one link of the graph. This is not a fault of OAuth.
Bearer tokens are easy to for clients to obtain, easy for clients to use for API calls and are widely used (with HTTPS) to secure public facing APIs from Google, Facebook, and many other services.