Is this your control? It is held to the boat by four screws (#3), and held together by two screws (#7) coming in from the backside. BUT you say yours is held to the boat by rivets? Those must be something substituted by the boat rigger guy.

Any Evinrude dealer can supply OEM cables, and any marine dealer can supply aftermarket cables (Sea Star, etc)

You should be aware that the cables design was changed in 1979. Not an issue unless you have a 1979-up control.

Teleflex 400 series cables (includes parts #51, 53 and 54 above) are excellent replacements at reasonable prices, but you need the plastic end-pieces from your old cable, or new pieces (Parts #41and the little parts that attach to the ends). They are sold in increments of one foot, and it’s important they are the right length. Several places sell them. If you have the style control box that is a few years older than the one shown above, the cables and related parts that hook to the motor are all the same. Below is a link to a place I have bought from:
Dave

https://www.boatersland.com/enginesystems-controlcables-tel400.html

• This reply was modified 4 years, 3 months ago by outbdnut2.
• This reply was modified 4 years, 3 months ago by outbdnut2.

Drill the heads off.