When your DSi reports that Wi-Fi access points are "locked," it's not a hardware fault—it's a service shutdown. The Nintendo Wi-Fi Connection (the backend that matched DSi consoles to online games, the DSi Shop, and friend services) was discontinued on May 20, 2014. The servers are gone; your DSi can still see Wi-Fi networks, but it has nowhere to connect through Nintendo's infrastructure.
Your DSi hardware is fine. The online ecosystem simply ceased to exist. For modern online Nintendo gaming, a 3DS, Switch, or Wii U is required.