Metro doesn't use PPP or directly public in any Indian city I can think of, they setup SPVs and actually have stable engineering and finance teams across contracts. And most of the engineers are taken from railways only in any case. And it's a really good promotion path, ministers are known to select successful metro spv administrators for lucrative roles in the state secretariat. They even have lateral movement between SPVs of different cities, e.g many top CMRL people are ex-DMRC. So the talent problem is not there.

The reason for delays are more boring: land acquisition, coordination among nhai, state pwd, railways, utilities, etc etc. But overwhelmingly land acquisition is the main bottleneck. If land acquisition fails or isn't exactly as you planned then you have to tweak the project itself which ripples delays all the way into the construction contracts, safety approvals, NOCs etc etc. After you resolve that, flyovers and roads are simultaneously being constructed in most cities since they are all expanding so you have to coordinate with that. And india from pre-independence has utilities placed under the middle of the road, as opposed to the sides of the road. Now this is not an iron clad rule (nothing is, in india) but it's generally true. This means that you also have to coordinate with utilities. And most of them were laid in the last century without any record left of where they were laid, so you can't even plan ahead you dig and you find out you've slashed a utility line. Each coordination point above is an NOC and all put together it takes time.