Can you log into the modem, navigate to the STATUS.....DOCSIS WAN page, copy the downstream and upstream tables and paste them into this thread. Those tables contain the RG-6 signal levels and signal to noise ratios which might show some type of problem. Was your previous ISP internet connection via DSL or Cable?
There is an ongoing issue with latency to and thru the CGN3 series modems which Hitron and Rogers are currently working on. Unfortunately there is no eta for the firmware fix.
The signal levels on the downstream are a little low, but that shouldn't cause any problems. The Signal to Noise levels are fine and the upstream signal levels are fine as well. Unfortunately you have the CGN3ACSMR, which is definitely having problems at this point in time. There are two issues on the go at the moment, 1. the latency to and thru the CGN3 series modems, 2. an issue specific to the CGN3ACSMR and League of Legends, where users either can't connect to the game or end up getting kicked off the game. The interesting thing is that Destiny is the game that is giving you the most problems. Does the xbox only ever use one particular set of ports for port forwarding or does that ever change according to the game? Are you able to easily determine that and if so, can you have a look at Destiny and a game that works and see if there are any differences in the ports that are used and post what you find.
One possible solution is to swap your modem for a CGN3 and see if that resolves the issue. The price for that however is giving up the 802.11ac capability and running into the problem with the CGN3 2.4 Ghz network, where it will lock down in data rate to the slowest device on the network, instead of running all devices at their rated wifi data rates, taking into consideration the signal to noise ratios of the wifi channel in use. On paper, the CGN3AC and presumeably the CGN3ACSMR run the 2.4 Ghz network as it should, allowing each device to run at its max wifi data rate available.
You mentioned that it seems to work in gateway mode so your issues are likely related to the NAT/bridging issues the other posters have mentioned. However, like you mentioned in your post Destiny is a game which has a reputation of having connectivity issues. I used to play Destiny on Xbox One a few months ago, and I experienced "the Noah's ark" of disconnect messages which you are probably familar with. If I remember correctly, my common disconnet errors were bee, fly, lion, and zebra. In the end I was able to solve my issues, so you might want to give this troubleshooting a try. Even though you are playing Destiny on Xbox 360, I think some of these troubleshooting steps should be of use to you.
What I discovered is Destiny is a game which is very sensitive to network latenency and packet loss. My issue was hard to track down as the issue would come and go. However, any time I would get disconnected I would see other issues in the house such as sluggish webpage browsing on my PC.
What I started doing was running a continous ping test on another computer in the house. In windows, if you go to Start > Run and type "ping -t www.google.com" it will bring up a command line window which will constantly ping google. When I would get disconnected from Destiny, I would go to the ping window on my computer and would see "request timed out" messages. While the ping isn't pinging the servers you are connecting to for Destiny, they can be used as a general indicator that you are receiving packet loss on your connection.
The root causes for my issue were two-fold:
1) I was connecting my Xbox One via wifi on the 2.4GHz spectrum. In the house we also had a baby monitor using the same frequency and would cause interference when it was on. You mentioned that you are wired, so this should not affect your Xbox 360.
2) When using computers with full wired connectivity, I would still experience packet loss at certain times of the day. To this day I don't fully understand why this was happening, but eventually Rogers sent a tech to my house to review the setup. What seemed to address the issue was connecting our cable feed to another card in the central box in our neighboorhood.
For what it's worth, I very recently switched over to the Ignite 250u plan using a Hitron CGN3. I played Destiny for about 5 minutes (not much of a test) with an out of the box setup and it seems to work fine.
Good luck with your issue. I hope this helps!
Do you use the wifi on the modem, and if so, just for test purposes, try turning it off and see if the max ping time comes down at all.
If you swap modems for something like the Cisco DPC-3825, that will limit your max download rate compared to what you can use with any of the Hitron modems. But, for test purposes, it will be interesting to see if the ping times come down at all. If not, then the issue is with the node that your modem runs off, and the network beyond it.