No, not at all. A raised SNR can be overcome in almost all circumstances by making more measurements, i.e. correlation, since noise is not correlated, it is removed. For the same reason random delays don't help against timing attacks.

Generally mitigations like adding random things only delay attacks like this, they don't prevent them.

Like adding random timings won't prevent timing attacks, adding random sized strings won't prevent chosen plaintext or padding attacks, etc...

Cryptography Research (now RAMBUS) developed solutions against these DPA attacks. Real solutions are much more complex than just "random-power-consuming"

See: https://www.rambus.com/security/dpa-countermeasures/