For a project like Signal, there are competing aspects of security:
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privacy and anonymity: keep as little identifiable information around as possible. This can be a life or death thing under repressive governments.
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safety and anti-abuse: reliably block bad actors such as spammers, and make it possible for users to reliably block specific people (e.g. a creepy stalker). This is really important for Signal to have a chance at mass appeal (which in turn makes it less suspicious to have Signal installed).
Phone number verification is the state of the art approach to make it more expensive for bad actors to create thousands of burner accounts, at the cost of preventing fully anonymous participation (depending on the difficulty of getting a prepaid SIM in your country).
Signal points out that sending verification SMS is actually one of its largest cost centers, currently accounting for 6M USD out of their 14M USD infrastructure budget: https://signal.org/blog/signal-is-expensive/
I’m sure they would be thrilled if there were cheaper anti-abuse measures.
There is no downside to nested encryption, except of course the performance overhead. But this only really makes sense if each layer has an independent key and each layer uses an algorithm from a different family. Improper key reuse weakens the scheme.
For symmetric cryptography like AES the benefit is dubious. It is far more likely that the content is decrypted because the key was acquired independently than that AES would be broken.
However, there absolutely is a benefit for asymmetric crypto and key agreement schemes. This is how current Post-Quantum Cryptography schemes work, because:
Nesting one algorithm from each family gives us the best of both worlds, at a performance overhead: conventional asymmetric cryptography give us temporary security in the near future, and the second PQC layer gives us a chance at long-term security.