Kentico's admin uses a lot of nested containers. If you inject your modal inside a deeply nested element, it might be constrained by overflow: hidden or z-index.
So, try - Render your modal at the end of the page body, not inside a container or web part layout.
Add this custom CSS:
.modal-backdrop {
z-index: 1060 !important;
}
.modal {
z-index: 1070 !important;
display: block; /* If needed for testing */
}
Note: Also, check that your custom styles are loaded after Kentico’s admin styles