Now you make me wonder if English speakers ever pronounce dept. literally.

Not often, but "dept" if sometimes used as a short form of "department" in written form, and I've heard people pronounce it literally before. It certainly can be pronounced distinctly from both "debt" and "depth" by native english speakers.

Yes, like that. Definitely not with a vowel. There's really no way to know for a non native speaker that the b is supposed to be silent, except by learning about it as a special case.

'pth' is different because the same stream of air can be used for both consonants, with the tongue contacting the teeth just as the lips open, so any gap between the consonants is not perceived to native listeners.

It'll probably sound like an accent to you, but then there are a thousand different native speaker accents, so that is OK with me.

For what it's worth, "bt" is not a common combination in my native Romanian either, but it is not hard to pronounce.

It can be done with lots of practice but native speakers don’t normally do this.