An active Wi-Fi icon does not settle which connection your Mac tries first. Apple documents a service order for multiple active network connections. You can inspect it in System Settings, Network, using the action menu and Set Service Order.
Separate three questions
Is the cable connected? Is its network service active? Is that service first in the order? Those are different checks. A familiar Wi-Fi icon alone cannot answer all three, and a cable attached to an adapter does not establish that the far end reaches the internet.
For a home-desk example, write down the existing order before changing anything. Identify the Ethernet service by its adapter name if necessary. If you deliberately place it above Wi-Fi, keep the previous order so you can reverse that choice.
Compare under ordinary conditions
- Confirm both services' current status.
- Avoid making changes during a call or file transfer.
- Open the same simple page before and after an agreed change.
- Record whether the symptom changed, rather than assuming the setting fixed it.
Service priority is not a promise that every application always uses an identical route. Nor does keeping both connections active automatically add their speeds together. On a managed Mac, ask the administrator before rearranging services. If Ethernet itself is slow, investigate that link separately from its place in the list.