A brief bridging vs routing discussion can be found here -> https://community.openvpn.net/openvpn/wiki/BridgingAndRouting
Is it as simple as just flipping the device mode from Tun to Tap on both ends?
The short answer is no. You also need to assign the tunnel to an interface and bridge that interface to your LAN. Then you need to coordinate the addressing of your devices, so there's no overlap. Not to mention, both sides will need to prevent their DHCP server's broadcast traffic from traversing the tunnel and causing issues on the other end.
In general, unless there is a specific requirement to access an application that only relies on broadcast traffic, a routed solution is your best bet.
My guess is Plex also supports some sort of direct IP mapping. In which case, I would stick with a routed solution. It will perform better, it's a simpler setup and it will only forward traffic that is destined for the remote end.