Uploading these tools as geoprocessing services fails

**Describe the bug**
We had a use-case where we wanted to use BatchTrace to propagate attributes downstream in our network. The testing we did in ArcGIS Pro was promising, but since the users will be working in a third-party solution connected to our ArcGIS Enterprise server we wanted to upload it as a geoprocessing service to enable its use through the ArcGIS Rest API. We followed the instructions to run the BatchTrace tool successfully and share it as a web service, but this fails with the following error message: "00068 Script Batch Trace contains broken project data source: String"

**To Reproduce**
Steps to reproduce the behavior:
1. Run BatchTrace tool successfully
2. Go to geoprocessing tool history
3. Press "Share as web tool"
4. fill in required fields, and press analyze
5. Analyze produces error "00068 Script Batch Trace contains broken project data source: String"

**Expected behavior**
Tool was made available as a geoprocessing service on my ArcGIS Enteprise server

**Desktop (please complete the following information):**
- OS: Windows 11, version 25H2
- ArcGIS Pro 3.5.2

Reproduce the BatchTrace workflow in ArcGIS Pro, then open geoprocessing tool history and use Share as web tool. Start with the Analyze result and error 00068, and trace the reported broken project data source. Done means the tool analyzes successfully and can be published as a geoprocessing service for the ArcGIS REST API.

Written by the indexing model from the issue text.