Right so JD really runs this as a USP - I'm not sure how different they really are than some other firms on this point, but it's obviously very important to them.
One thing here is structural - JD operates a unitary partnership, including profit share across borders. Cf.
Baker McKenzie, which is similarly globally distributed but operates as a Swiss Verein, with profits siloed by operating entity (usually per jurisdiction).
What proponents of unitary Partnership will tell you is that their model is better because it means that the team in, say, Singapore is fully incentivised to give their all on an instruction originating in, say, Paris. The criticism of the Verein model by contrast is that there is an incentive in each office to prioritise work originated there over work brought in from other offices. Proponents of the Verein will say this problem is overstated, that there's a global commitment to the brand, and the offices aren't siloed for operational purposes. All of this is debatable - JD obviously leans into the unitary model so think through those advantages.
The 'no home office' component follows a related line of argument. There's a perception that the global firms which have grown out of the US and UK are still 'domiciled' in their home country, that they're best suited to work originating in the home country or to be conducted in that home country. This is obviously somewhat true for like, Slaughters and Wachtell, which have minimal global footprints. It is arguably somewhat true for the American PE heavyweights/spec firms, simply because a huge amount of PE work is American capital flowing into int'l businesses - although the UK, Europe and Asia all have their own thriving PE industries, American investment firms loom large.
Firms like JD, W&C, Bakers, (the list goes on) will say 'it doesn't matter where we came from, what matters is where we are and where we're going' - they do not want to be perceived as American firms. This is partly about projecting global competence (we can deal with your problem in any jurisdiction) and partly about appealing to global clients (we are not a local office of a foreign outfit, we are as much part of the legal market of [country] as [notable domestic firm].'