Possibly another stupid question from me here, but can the Lightning Network then be leveraged as a use case for tumbling/mixing/hiding our UTXOs after the upgrade?
It's already possible. The real advantage of Taproot on LN is hiding other condition of the HTLC which isn't executed.
The only downside is that you must leave the channel reserve in the old channel.
Workflow is as follows:
1. Open 2 Lightning nodes; one with outgoing liquidity and the other with an equal amount of incoming liquidity (minus the reserve)
2. Send everything from the node with 100% outgoing to the 100% incoming node
You just did an ad-hoc Coinswap over lightning.
n.b. I think eltoo channels remove the reserve requirement, so that would mean you could perform this without the "loose end" constituted by the reserve balance left over on one node
However, a 'real' coinswap is better. It's probably less traceable (at least than HTLC based Coinswap-over-LN) and the LN technnique might spark a trend of people closing both channels and nodes once they finish the swap (which is a little anti-social, payment channels are intended to be long lived).