Swift UI - Using UIVIewControllerRepresentable to update logs

You haven't provided a Minimal Reproducible Example but here is a simplified version of what seems to be what you are trying to do.

First, add a LogManager that can be created by ANY class or struct

struct LogManager{
    var name: String
    ///Simplified post that takes in the String and uses the name as the source
    func postMessage(message: String){
        postMessage(message: .init(timestamp: Date(), message: message, source: name))
    }
    //MARK: Notification
    ///Sends a Notification with the provided message
    func postMessage(message: Message){
        NotificationCenter.default.post(name: .logManagerMessage, object: message)
    }
    ///Adds an observer to the manager's notification
    func observeMessageNotification(observer: Any, selector: Selector){
        NotificationCenter.default.addObserver(observer, selector: selector, name: .logManagerMessage, object: nil)
    }
}

Put at class or struct level the declaration for the manager

private let log = LogManager(name: "YourClassStructName")

Then call

log.postMessage(message: "your message here")

When you want to log a message.

In the ViewModel you would

1. subscribe to the notifications
2. maintain the array

Like below

class AppLogSampleViewModel: ObservableObject{
    static let shared: AppLogSampleViewModel = .init()
    private let manager = LogManager(name: "AppLogSampleViewModel")
    @Published var messages: [Message] = []
    private init(){
        //Observe the manager's notification
        manager.observeMessageNotification(observer: self, selector: #selector(postMessage(notification:)))
    }
    ///Puts the messages received into the array
    @objc
    func postMessage(notification: Notification){
        if notification.object is Message{
            messages.append(notification.object as! Message)
        }else{
            messages.append(.init(timestamp: Date(), message: "Notification received did not have message", source: "AppLogSampleViewModel :: \(#function)"))
        }
    }
} 

If your View won't be at the highest level you need to call.

let startListening: AppLogSampleViewModel = .shared

In the ContentView or AppDelegate so the ViewModel starts listening as quickly as possible. You won't necessarily use it for anything but it needs to be called as soon as you want it to start logging messages.

Only the View that shows the messages uses the instance of the ViewModel.

struct AppLogSampleView: View {
    @StateObject var vm: AppLogSampleViewModel = .shared
    //Create this variable anywhere in your app
    private let log = LogManager(name: "AppLogSampleView")
    var body: some View {
        List{
            Button("post", action: {
                //Post like this anywhere in your app
                log.postMessage(message: "random from button")
            })
            DisclosureGroup("log messages"){
                ForEach(vm.messages, content: { message in
                    VStack{
                        Text(message.timestamp.description)
                        Text(message.message)
                        Text(message.source)
                    }
                })
            }
        }
    }
}

Here is the rest of the code you need to get this sample working.

struct AppLogSampleView_Previews: PreviewProvider {
    static var previews: some View {
        AppLogSampleView()
    }
}
extension Notification.Name {
    static let logManagerMessage = Notification.Name("logManagerMessage")
}
struct Message: Identifiable{
    let id: UUID = .init()
    var timestamp: Date
    var message: String
    var source: String
}

Only the one View that shows the messages needs access to the ViewModel and only one ViewModel subscribes to the notifications.

All the other classes and structs will just create an instance of the manager and call the method to post the messages.

There is no sharing/passing between the classes and structs everybody gets their own instance.

Your manager can have as many or as little methods as you want, mine usually mimics the Logger from osLog with log, info, debug and error methods.

The specific methods call the osLog and 3rd party Analytics Services corresponding methods.

Also, the error method sends a notification that a ViewModel at the top level receives and shows an Alert.

To get all this detail working it takes a lot more code but you have the pattern with the code above.

In your code, in the updateUIViewController you break the single source if truth rule by copying the messages and putting them in another source of truth, right here.

fullLogMessage = logViewModel.getTheFullLogMessage()

This is also done without a check to make sure that you don't go into an infinite loop. Anytime there is code in an update method you should check that the work actually needs to be done. Such as comparing that the new location doesn't already match the old location.