Appchains for Sovereignty
Verdict: The Clear Choice. Appchains (e.g., Cosmos SDK, Polygon Supernets, Avalanche Subnets) are purpose-built for teams requiring maximum control. You own the validator set, can implement custom fee tokens, and fork the chain's code without external governance. This is critical for protocols like dYdX (migrated to a Cosmos appchain) that need to tailor throughput, MEV policies, and upgrade schedules.
Key Trade-off: You are responsible for your own security and validator bootstrapping. The blast radius of a bug or exploit is confined to your chain, but so is the security budget.
Rollups for Sovereignty
Verdict: Limited. Rollups (e.g., Arbitrum Orbit, OP Stack, zkSync Hyperchains) inherit significant constraints from their parent L1 (Ethereum). While you can customize gas tokens and precompiles, you cannot modify core consensus or data availability rules. Your chain's liveness depends on the rollup sequencer and the parent chain's health.