As I understand it, the DL and EY planes were leased months ago and have already been delivered or parked somewhere waiting to be delivered.

You don't need more pilots and crew to operate the same routes with newer planes. You might certainly need training, but these leased planes are 777s. How much retraining is needed?

Clearly there is a roadblock somewhere. But where? Maybe there is not enough orange paint in the world to put the AI logo on the exteriors...supply chain problem. But my preferred theory is that the AI reservations system cannot handle the new seat layouts, and has to be rewritten from scratch. Yes, I am sure that's it.

4 out 5 DL planes are operating and the 5th one is yet to be delivered. Not a single one of the 6 EY planes have been delivered yet.

Followup question: what does "delivered" mean? Is it exactly the same as "not operating" (which we already know)? Why are they sitting somewhere without being delivered? We know they are not being used by DL or EY.

They are coming out of storage. So being prepared for service (both cabin fixes and technical checks). There is no conspiracy here and no plane just sitting there. AI has done a great job IMHO of getting these leased planes in a market that is very tight on aircraft. The EY planes are all being prepped as well. The Aeroflot Airbus planes are coming off the Airbus line. I believe one aircraft is being painted.

AI has a severe crew shortage in the B777 fleet, so even if the aircraft was available you still need crews. Each widebody requires about 300 direct employees at an airline, one widebody requires about 10 senior pilots and 20 junior pilots.

Aircraft that were stored need a number of checks to be ready for service. All the MRO shops worldwide are currently blocked with such work. Then comes the recertification of the aircraft to be registered under the Indian registry.
An example: LH leased 5 A359 which used to fly with PR in July 2021. The first aircraft was put into service only on April 29, 2022 and the remaining 4 came in every 3-4 weeks till August 22. And this after LH cut a lot of corners: interiors were mostly left as is and exterior on one aircraft was not painted with only the PR decals removed. They needed the capacity for the busy summer months. In the slower winter season 22/23 the aircraft went back into the shop for modification to fix a number of seats and remove/rearrange some interiors to fit more seats in PE.
Another example: reactivating their own A346 from deep storage took them 2 years for 10 aircraft. Those required no regulatory paper work, no paint work, just activation in TEV (2-3 weeks), C check at LHT in MNL and some work in FRA/MUC.

AI has no pull with various MRO/paint shops unlike LH who owns LHT. So one year to get the EY aircraft into service sounds about right ;)

I believe they are getting it done at SQ's MRO which makes a lot of sense considering their stake in AI.

Biggest complaints, I see based on my travel and from other fellow passengers are the IFE (both business and economy class) and some broken seats (recline etc.). IFE is important on long haul flights (7 hrs+) and someone paying top dollar in business wants IFE to be working and have some variety. Dont think it is that difficult to fix this and the broken seats. Dont need to refurbish the entire plane to fix the IFE and seats. Plus points are - Food is excellent (at least the Indian food), baggage handling quite good, timings and on time arrival is adequate too.

Swapping out the installed Thales system is not an easy task. The IFE display and remote are certified together with the seat, if you change the setup you need to recertify the entire setup. Broken seats should be really marked as such and taken out of sale. To change on row in the middle of the aircraft you need to take the seats ahead or behind out, so it is not so easily done during a A or S check. Currently every available widebody is in service to service the very lucrative US-India market, each B777 represents 5-6% of capacity. I seriously doubt they will take 5-10% of their capacity out right now to fix things that are not revenue relevant. Once the EY B77L and new A359 capacity comes in they will perhaps think of that or straightawy go for the tip to tail refurb planned to start in W23/24 anyway.