No. Lightning payments are not public. The receiver, and all the intermediary nodes, cannot know from where the payment came. There are also private channels that you can open to some nodes. They will be invisible to outside observers. So it is very easy to pay for goods and services anonymously. Receiving is a bit harder to arrange, but there are blinded paths. This tech is mature and quite reliable. 

Shielded Bitcoin solution is new and untested. You would still have to run some extra software to use it, and there can be bugs and exploits. To me it looks like another attempt at RGB.

Also, the BTC you peg out will be treated the same as coming out of a mixer. Any chain analysis/AML filter will flag them as "dirty".

SwapMarket

They are different privacy models. Might even be complimentary.

As I understand it, the ability to spend from arbitrary peg-in vaults adds a fungibility that Lightning cannot offer, as in Lightning, value must transfer between the established channels on a node, which is public.

bitcoin

privacy

Shielded Bitcoin: Private Transfers on Bitcoin L1

Kruw

They are different privacy models. Might even be complimentary.

As I understand it, the ability to spend from arbitrary peg-in vaults adds a fungibility that Lightning cannot offer, as in Lightning, value must transfer between the established channels on a node, which is public.