I'm not sure what's wrong, but NTLite works fine for me. KB5014676 (June 2022) is the current Dynamic Update for W10 21H2.
After downloading, Updates folder should have:
Directory of C:\Program Files\NTLite\Updates\10.21H2.x64
08/09/2022 08:48 AM 70,518,551 windows10.0-kb5013887-x64-ndp48_cee9944aef2e07c7ca85b5ca1d9d67ecdf7a9abf.msu
08/09/2022 07:53 AM 15,428,866 windows10.0-kb5014676-x64_e054ac3346896e28d2cae68acdf899c0ee274f11.cab
08/09/2022 08:55 AM 727,621,123 windows10.0-kb5015807-x64_e3e62de3b52802c48fa6b027c49e0951b8575743.msu
If you're still having integration errors, then attach your preset, NTLite.log, and NTLite_dism.log (from AppData/Local/Temp).