Definitely a lot of nuance and caveats... Lightning receiver privacy has always been bad. Amount privacy also not guaranteed because of probing like you said.

Self custody Lightning can be more/less private than on-chain depending "how" (with or without LSP) you use it and "who" you want to be private from (counter parties, third parties, or both)

i.e.
Using unannounced channels gives privacy from third parties (general public)...but not counter parties (the user youre transacting with) and ironically destroys plausible deniability.
Wrapped invoices with Zeus gives receiver privacy from counter parties, but not third parties (Zeus).

How most use lightning (with custodians and LSPs 80-90% according to nostr and wallet downloads) it offer similar privacy to a traditional bank. It gives privacy from the general public, but not the custodian/LSP.

Noticed you are using Wallet of Satoshi for Nostr. If you're using lightning custodially (so you've already accepted the risk of being rugged), and you are concerned about privacy, you should consider using Ecash for that since it is a strict improvement. Privacy from counter parties AND third parties.

https://zapalytics.com/