I’ve lived in 4 different flats in the Barbican and they all had a dishwasher. I think only the studios you’d have a problem finding space for one. Of course in the others it is a preference whether you want to lose space for other things or not. It is not a lot of extra plumbing. It is usually when they want to preserve the original kitchen (or a cheap landlord as you suggest - although all the ones I had there were great)
That’s on the cheap end of zone 1 London for service charges, especially anything with a porter and gardeners. If you were buying now you’d renew the lease for about £20k, which is a tiny fraction of the purchase price you’d be paying. (Source: did it in 2022)
It also depends where you are in the Barbican. It spans between the Barbican station and Moorgate. If you are at the latter you can enter Elizabeth line from there too.