You should be able to see your archives; as far as I know, even basic can see their archive on the default site.
As to analysis, you have to run that and then save it if you would like to review it later and don't want to re-run (a recently added feature -- https://www.chess.com/blog/News/chess-com-dev-update-june-9-2017 ).
If you are not seeing your archive, something is wrong. How are you connecting to the site and trying to look at your archive?
I pay for premium membership and I can' see archive games and not see analyse. This is the reason why people pay for premium membership at chess.com. So, don't pay! It is stupid decision. Save your money and do not pay any membership at chess.com!