You should require payment and performance bonds from your subcontractors, especially for large portions of the contract value like electrical and mechanical work, because they protect you from subcontractor default.
Your prime bond covers your obligations to the county, but it does not cover the risk of your subs failing to complete their work or pay their own suppliers and laborers. If a sub defaults, you as the prime are still on the hook to the county for that work, and a sub bond shifts that risk to the sub's surety. This protects your cash flow and reputation. The Miller Act requires prime contractors on federal construction projects over $150,000 to provide performance and payment bonds, but it does not directly address sub-tier bonding. State and local requirements vary, so check your county's procurement code and your contract. Your surety likely expects you to mitigate risk where possible, and sub bonds are a key way to do that on larger projects. The cost of sub bonds is typically passed through to the project owner as part of the subcontractor's bid, similar to how your prime bond cost is part of your bid to the county.