Ok. Perfect. Now I know what happened. Mine is a very old computer. That may explain the timeout.
It's not much related to the PC's performance. Armory's p2p broadcast code was implemented when Core would push transactions as is once delivered by another peer. Core batches now, so the broadcast detection throws a lot of false negatives. This needs a bit of a technical explanation to make sense:
Early days, before Core was even named Core, the Bitcoin node software did 2 things different: Setting up the JSON-RPC interface required some minor work on the user part, notably adding login and password to a config file. Therefor, Armory could not expect access to the RPC in plug and play setups (95% of users), so it had to rely on something else to broadcast transactions. The RPC returns with an ACK/nACK on broadcasts but if only 5% of users can set it up properly, it was useless to Armory.
Instead etotheipi (the original maintainer) opted to connect to Core through the P2P interface. Armory, still to this day, spoofs a P2P node to your Core instance, for the purpose of pushing transactions and getting new block notifications.
The P2P broadcast interface does not come with an explicit ACK/nACK. It doesn't provide a result at all as a matter of fact, for design reasons. The flow goes like this: advertise the hash of the transaction you want to broadcast to the peer. The peer will then request that transaction by hash. You will return the transaction body to the peer. Broadcast is over. The only way to know if that transaction made it into the peer's mempool is to query that peer for the same transaction. If it returns it to you, it's in. You won't get a reply otherwise.
So basically, Armory sends the transaction to your Core instance and polls it for a few seconds to verify it's been added to the mempool. Eventually core introduced broadcast batching, where it tries to make a better use of the MTU and reduce point of origin timing leaks. Basically your node will now wait some time and try to accumulate a bunch of transactions before announcing them to other peers. This delay also affects entry in the mempool (so that you can't just query the node for specific hashes to know what made it in or not with certainty). As a result, this disrupts the Armory P2P broadcast loop, since the polling may now timeout.
Since then, interfacing with the RPC has become easier so I've added a RPC broadcast fallback when available, which is why you received the message about the RPC.