Can I troubleshoot fiber converter with wireshark "unexposed"?

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Hi! My ISP is causing me some trouble which I hope wireshark might help at least to diagnose..

Background Setup: Fiber connection - fiber converter - wiring - router - devices of all kinds Three different routers randomly restarts or freezes (D-Link DIR-655, Asus RT N56U both become unresponsive, D-Link DIR-882 restarts at least some internal interface) 5-20 times per day. An ethernet cable "tester" shows that the wiring to the router should be fine, and the problem is persistent even with the router connected directly to the fiber converter via a short brand new ethernet cable.

Now my ISPs second line support wants me to connect a computer directly to the fiber converter to omit the router as the problem. I take a bit of an issue with not having the router as an extra point of security I don't really like that suggestion, never mind the fact that they actually seem to think that I should personally monitor the connectivity of the computer, while checking if "the lights go out" on the fiber converter to see if this random problem (could occur with 5 minute intervals or 12 h...) occurs even with that setup.

Question: Would it make sense to connect a computer running wireshark to the fiber converter (has a built in switch), without accepting an IP adress assignment to monitor the status of the fiber converter/connection itself? (I guess listening to the chatter basically to see if it dies out..)