You do not say where you "read that ArcObjects is bundled with the Desktop license" but with respect to your question about the ArcGIS for Home Use Program:
This annual subscription gets you:
- ArcGIS for Desktop Advanced (ArcMap and ArcGIS Pro)
- The most popular ArcGIS for Desktop extensions
- ArcGIS for Desktop software updates
- ArcGIS Online Named User Account
- 100 service credits for data storage, premium data access, and geocoding and analysis
- A suite of ready-to-use apps for use anywhere, on any device
- A Living Atlas of the World with maps and data, including access to foundation content from Esri, on thousands of topics
- Installation support
There is no mention of the ArcObjects SDK for .NET being included, and from Downloading ArcObjects SDKs for .NET? I would not expect it to be:
the ArcObjects SDK (.NET or Java) ... is a licensed product
Consequently, I think you should contact Esri to be certain of your options.