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Is it possible in Monero to have non-deterministic wallets (setting aside ergonomic issues such as ease of setting up the wallet)? I.e., if I use two random numbers as private spend and view keys, will they constitute a useable but different wallet than the same private spend key paired with its derived private view key?

If the answer to the above is positive, does one lose any functionality by using such a wallet?

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Originally Monero used non-deterministic wallets. Deterministic (and mnemonics) are a convenience feature.

Take a look at the --non-deterministic command-line flag, which will generate a non-deterministic wallet, and also --restore-from-keys flag, which will restore a non-deterministic wallet (by prompting for the view and spend keys).

No functionality is lost (beyond any ease-of-use issues).

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