Yes. Your first point is something that I would happily engage in as well. I have no problems contributing to the community. Moreover, the extension can have several additional options like:

(i) Do not perform any kind of inference on the client, i.e. always use query existing timestamps from an the online database. This will be helpful for users with low power devices like laptops.

or

(ii) Perform inference (only) for the video that the client wants. This is, of course, necessary if the video does not have any timestamps on the server. It does the inference and uploads the results on the central server.

or

(iii) Keep performing inference for new videos (even ones that are not watched by the particular user) - Some folks who runs a powerful enough hardware and are eager to donate their computation time can choose this option. I am pretty sure some folks will emerge who are willing to do this. The LeelaChessZero project banked entirely on this particular idea. For this option, there could be slider to let the user control how much of the resources to keep actively engaged (maybe by limiting thread count).

The second point that you mentioned could be a implemented with a peer-to-peer communication protocol, but if the neural network's weights don't change, then there would be nothing different with most recent vs. stale timestamps. Also, in P2P you'd still need trackers to keep track of peers, which could be a central server or be decentralized and serverless depending on the implementation. One potential problem could be latency though.