The error is:

Passing argument of non-sendable type ‘AVAssetExportSession’ outside of main actor-isolated context may introduce data races; this is an error in the Swift 6 language mode

This is telling you that the problem is arising because you are performing this in a main actor-isolated function.

The easy fix is to remove this actor-isolation, to make the function nonisolated:

nonisolated func extractAudioFromImportedVideo(videoURL: URL, success: @escaping ((URL) -> Void), failure: @escaping ((Error?) -> Void)) async {
    …
}

Personally, I would eliminate these closures and just follow async patterns:

nonisolated func extractAudioFromImportedVideo(videoURL: URL) async throws -> URL {
    do {
        let asset = AVURLAsset(url: videoURL)
        let audioTracks = try await asset.loadTracks(withMediaType: .audio)

        guard !audioTracks.isEmpty else {
            throw AudioExtractionError.noAudio
        }

        guard let exportSession = AVAssetExportSession(asset: asset, presetName: AVAssetExportPresetAppleM4A) else {
            throw AudioExtractionError.exportFailed
        }

        guard let audioURL = makeFileOutputURL(fileName: "ExtractedAudio.m4a") else {
            throw AudioExtractionError.outputUrlFailed
        }

        print("audioTracks", audioTracks)
        exportSession.outputURL = audioURL
        exportSession.outputFileType = .m4a


        await exportSession.export()
        print("exportSession.status", exportSession.status.rawValue)
        if let error = exportSession.error {
            throw error
        }

        print("Audio extraction succeeded")
        return audioURL
    } catch {
        print("extractAudioFromImportedVideo Error:", error.localizedDescription)
        throw error
    }
}

Note, I also retired NSError, replacing it with:

enum AudioExtractionError: LocalizedError {
    case noAudio
    case exportFailed
    case outputUrlFailed

    nonisolated var errorDescription: String? {
        return switch self {
        case .noAudio: String(localized: "Video does not contain audio", comment: "AudioExtractionError")
        case .exportFailed: String(localized: "Failed to create export session", comment: "AudioExtractionError")
        case .outputUrlFailed: String(localized: "Failed to create output URL", comment: "AudioExtractionError")
        }
    }
}

That is actually localized, isolates the details from this function, and enumerated errors support richer error handling.

FWIW, once you have this working, you can simplify further:

nonisolated func extractAudioFromImportedVideo(videoURL: URL) async throws -> URL {
    let asset = AVURLAsset(url: videoURL)
    let audioTracks = try await asset.loadTracks(withMediaType: .audio)

    guard !audioTracks.isEmpty else {
        throw AudioExtractionError.noAudio
    }

    guard let exportSession = AVAssetExportSession(asset: asset, presetName: AVAssetExportPresetAppleM4A) else {
        throw AudioExtractionError.exportFailed
    }

    guard let audioURL = makeFileOutputURL(fileName: "ExtractedAudio.m4a") else {
        throw AudioExtractionError.outputUrlFailed
    }

    exportSession.outputURL = audioURL
    exportSession.outputFileType = .m4a

    await exportSession.export()
    if let error = exportSession.error {
        throw error
    }

    return audioURL
}