You have no idea what lightning transaction volume is, for all you can tell I'm currently making 1000 transactions per second in a loop between a set of lightning wallets. :)
Because lightning is actually relatively scalable it doesn't broadcast every action to everyone.
The current implementation of LN does broadcast to everyone. At any given moment, every node on the network needs to know the size of every open channel on the network (for routing calculations) so every time a transaction happens, the updated channels sizes get broadcast to everyone. The volume is almost zero right now.
That isn't so. You learn that a channel doesn't have enough capacity by attempting to use it, failing, then updating your knowledge and trying again. It typically takes something like 1.25 tries on average in the current network.
See Bolt 7:
> Note that the htlc_maximum_msat field is static in the current protocol over the life of the channel: it is not designed to be indicative of real-time channel capacity in each direction, which would be both a massive data leak and uselessly spam the network