When doing transcription of English (British or American) in IPA, is broad transcription exactly the kind of transcription which doesn't make use diacritics, and narrow transcription the kind which makes use of diacritics?
First off, "broad" versus "narrow" transcription is a spectrum. One transcription can be broader or narrower than another (or broader/narrower in particular ways) but there's no firm dividing line that makes some transcriptions "broad" and others "narrow" in a vacuum.
But also, a very broad transcription can use diacritics: the Kota language has a three-way contrast between dental, alveolar, and retroflex stops, and the IPA doesn't have three separate symbols for those, so any useful IPA transcription (no matter how broad) will need to find another way to distinguish them. Outside of IPA, it's common to use symbols with diacritics like
č in broad transcriptions.
Similarly, a very narrow transcription doesn't always need diacritics. If you're transcribing a perfect exemplar of a high front unrounded vowel in IPA, then that's simply
[i], no diacritics needed.
For English, that is a reasonable standard. There are languages with phonemic distinctions ("broad transcription") that require diacritics such as the dental vs alveolar distinction, or sonorant phonatory distinctions like [a̤] versus [a], or [m] vs [m̥]. Any such differences are part of narrow transcription in English. You could decide that [i] is really "advancd i" so you could systematically write [ɪ̟] instead of [i] (English "i" is far enough off from the IPA reference value that this is viable), but the desideratum of only using diacritics when unavoidable rules against that choice, therefore broad transcriptions should be as diacritic-free as possible. More accurately, the broadest transcriptions exclude all sub-phonemic detail, then you can include more and more sub-phonemic detail is an increasingly narrow transcription, which will at some point include diacritics and also auxiliary (often phonemic – but general-purpose) symbols like [ʲ ʰ].