So here are some observations…
192.168.1.1 is handled by WiFi, not wired. Same for 192.168.1.176. 192.168.137.x addresses are handled by wired. The addresses you are using do not correspond to the highest priority route (the route with lowest metric…eth0 is wired and lowest metric, wlan0 is wireless and uses the lower priority/highest metric).
Addresses falling outside of those subnets would be redirected first through wired, and if wired is down, switch to WiFi. Something seems off that you are logging in through the IP address which is the highest metric…as if eth0 has failed. But metric on your host PC might differ than what the Jetson is using.
It appears wired is up, but you still need to post your traceroute results to really verify which interface things are going through, especially for “traceroute 192.168.137.1”. It would be nice to see if the calls to the wired route go straight there, or if instead they try to go to WiFi in order to reach wired (which would be wrong). It would also be good to find out what “dsldevice.lan” is by seeing what traceroute says about it since this is the name given to the higher metric WiFi’s default route. This seems like something is off in configuration. Look again at the traceroute questions in #8:
I would actually suggest completely shutting down WiFi to test with:
sudo ifdown wlan0
# Test chromium...
# ...and then test again with only eth0:
sudo ifup wlan0
sudo ifdown eth0
# ...test chromium again...