The reason to re-encrypt and inject the bridged messages into Matrix is that they are then persisted serverside in a standard e2ee manner, which you can then get at from any client that speaks Matrix (or is bridged in turn to Matrix) without that client or bridge having to know anything about the original network.

So, in pidgin, you are stuck with pidgin’s UX whether you like it or not in order to access the remote network (unless you use a different multihead client). Whereas with a matrix bridge, running either serverside or clientside, you can then connect to the bridged conversations using any UX you like (of which there are now hundreds of Matrix apps of different flavours). Moreover, each chat’s history gets decentralised across any other Matrix servers whose users are participating in that conversation.

The architecture is more like bitlbee, for those who know it - but using Matrix rather than IRC as the protocol between the client and bitlbee, so the richness of the comms is preserved (rather than flattening everything to IRC).