I'm looking to eliminate the adverb in "as X will explain gladly". Which word would be suitable here?
Generally in a presentation a native speaker will politely indicate that someone will be glad to explain by saying something like we welcome your questions. To be closer to the format in your original question, you could modify this phrase to X will welcome questions or whatever precisely fits your intended meaning.
Another common choice is appreciate as in we appreciate your questions.
The native style often translates something that means happy into something that means anticipates and will generally use the word questions instead of explain (but creating the same meaning) which results in a sort of softer style that you seem to be searching for and which is very common in business. So one can also hear X looks forward to your questions or X is eager to answer your questions or something along those lines.