Do hierarchical deterministic wallets require higher fees for privacy benefits?

You are correct that spending multiple UTXOs increases the transaction size and hence the fees. However, because each UTXO is individually signed, it does not matter whether they all belong to the same address or to different addresses, so there is no benefit to address reuse.

What you can do is consolidate small UTXOs into larger UTXOs during times when transaction fees are low, but be mindful of the privacy implications of this.