Service Principal Credential Configuration Error: PowerBiNotAuthorized

Here is the script that finally worked for me. I was suggested to get the OAuth2 access token in some other way and I did it that way. I am not sure how to tell from the power bi service if the data source is actually authenticated. However, I am assuming that it is authenticated because the script does not output an error. 

# Set execution policy for Task Scheduler compatibility
Set-ExecutionPolicy -Scope Process -ExecutionPolicy Bypass

$tenantId = ''
$vaultName = ""
$secretName = ""
$applicationId = ""

# Power BI Application Authentication
$secret = Get-AzKeyVaultSecret -VaultName $vaultName -Name $secretName -AsPlainText
$securePassword = $secret | ConvertTo-SecureString -AsPlainText -Force
$credential = New-Object -TypeName System.Management.Automation.PSCredential -ArgumentList $applicationId, $securePassword
Connect-PowerBIServiceAccount -ServicePrincipal -Credential $credential -TenantId $tenantId

# Get PAT (through OAuth2 client credentials flow)
$clientId = $applicationId
$clientSecret = $secret
$scope = "2ff814a6-3304-4ab8-85cb-cd0e6f879c1d/.default"

$body = @{
    client_id     = $clientId
    scope         = $scope
    client_secret = $clientSecret
    grant_type    = "client_credentials"
}

$response = Invoke-RestMethod -Method Post `
    -Uri "https://login.microsoftonline.com/$tenantId/oauth2/v2.0/token" `
    -ContentType "application/x-www-form-urlencoded" `
    -Body $body

$accessToken = $response.access_token

# ---------------- MAIN SCRIPT ----------------

$datasetId = ""
$workspaceId = ""

$datasources = Get-PowerBIDatasource -DatasetId $datasetId -WorkspaceId $workspaceId
$datasource = $datasources[0]

# # if no gateway, stop
if (-not $datasource.GatewayId -or -not $datasource.DatasourceId) {
    Write-Host "No datasource found or no gateway associated with the datasource. Exiting script."
    return
}

$gatewayId = $datasource.GatewayId
$datasourceId = $datasource.DatasourceId

# Take Over
Invoke-PowerBIRestMethod -Url ("https://api.powerbi.com/v1.0/myorg/groups/$workspaceId/datasets/$datasetId/Default.TakeOver") -Method Post -ErrorAction Stop

# Bind to Gateway (because after taking over, the dataset is not bound to the same gateway)
$body = @{
    "gatewayObjectId"     = $gatewayId
    "datasourceObjectIds" = @($datasourceId)
} | ConvertTo-Json -Depth 10


Invoke-PowerBIRestMethod -Url ("https://api.powerbi.com/v1.0/myorg/groups/$workspaceId/datasets/$datasetId/Default.BindToGateway") `
    -Method Post `
    -Body $body `
    -ContentType "application/json" `
    -ErrorAction Stop

# Reget the datasource after taking over and binding to gateway (this is necessary because the datasource might change after these operations)
$datasources = Get-PowerBIDatasource -DatasetId $datasetId -WorkspaceId $workspaceId
$datasource = $datasources[0]

$datasourceId = $datasource.DatasourceId
$gatewayId = $datasource.GatewayId

$body = @{
    credentialDetails = @{
        credentials                 = "{""credentialData"": [{""name"": ""accessToken"", ""value"": ""$accessToken""}]}"
        credentialType              = "OAuth2"
        encryptedConnection         = "Encrypted"
        encryptionAlgorithm         = "None"
        privacyLevel                = "Public"
        useEndUserOAuth2Credentials = $false
        useCallerAADIdentity        = $false
    }
} |  ConvertTo-Json -Depth 10

# Update the datasource with the new credentials
Invoke-PowerBIRestMethod -Url ("https://api.powerbi.com/v1.0/myorg/gateways/$($gatewayId)/datasources/$($datasourceId)") -Method PATCH -Body $body -ErrorAction Stop -ContentType "application/json"