Whether a safari, desert journey or a hike to see the magnificent mountain gorillas of Rwanda, all our itineraries provide endless fascination.

What is this phrase in bold and how does it
function/what does it modify?

Would you say it is short for

a.Regardless of whether it be a safari...
b.Whether it be a safari...
d. Not short for anything

It's a verbless clause: a or b or 'whether it is...'. I can only suppose it complements 'itineraries'.
Why do you think they have reduced it?

Do you think it is natural to write it or say it thus?
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No one has 'reduced it'. Verbless clauses are a normal mode of expression.
What I'm saying is that I feel like the version where bits have been left out (to avoid using 'reduced') is uncommon in speech, unlike the version with the bits included...

So that's why it seems as though the version is reduced from the common, unreduced version...

Still wrong? The reduced version is common in speech and writing??
Well, 'whether' clauses like any of those are uncommon in speech, I think, but 'whether it be' is surely less so! I see no reason either to suppose that writing reduces spoken language. Right?
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the clause in bold is indeed a verbless clause. The process of its formation is known as syntactic compression, so this may coincide with what you mean by 'short'. We could also call it reduction.

Concerning the syntactic function: the clause in bold is a conditional-concessive clause (whether... or usually introduces them).

The verbless clause is usually said to be the reduction of Subject + BE (in the indicative mood), but since subjunctive is possible in such constructions (although becoming increasingly uncommon), option 'b' seems correct.

Respectfully, Gleb Chebrikoff
Hi, Glen, and thank you.

Could we not argue option a? A and b seem to me to mean 'No mattter whether it be this or that, ...'