The plane is too big for 2 engines. But the massive engines for 2-engined planes are too big and inefficient for use in a 4 engine configuration. For example, each engine on 2-engine plane must be powerful enough alone for takeoff, whereas in a 4-engine configuration safety regulations only require a minimum of 2 (or 3?) working engines. The former require a much larger performance envelope, even though that increased performance is only required for a brief moment if ever, which is a source of inefficiency. Similarly, maintenance requirements are stricter for the former as they require more frequent safety checks and because they're run harder during takeoff (to minimize inefficiency from the wider performance profile) they break down more frequently.

This is why a 4-engined plane like the A380 isn't necessarily less economical for having 4 engines instead of 2. Airbus and Emirates have always argued, and arguably Emirates has demonstrated, that all things considered the A380's 4 engine configuration is still cost competitive. However, these factors cannot overcome the aged engine technology. When the A380 debuted, its engines were already half a generation behind the latest, greatest wide body engines, and now they're even further behind. Moreover, IIRC, even older 777 models have received engines upgrades so now the A380's engines are technologically behind aircraft that preceded it.

Airbus and Emirates weren't asking Rolls Royce to deliver anything fancy; just a replacement engine sporting updated blade designs, improved bypass, and other ubiquitous advancements since the mid 1990s; but nothing bleeding edge, like new gearing technology.

Hat tip for mentioning "gearing technology". The Pratt & Whitney PW1000G engine is just wild to me. It is a huge leap forward (no pun intended, as CFM's latest is called LEAP). Like all major redesigns, there will be years of bugs and minor failures to work through. Fortunately, they are starting much earlier than all the competition.

EDIT (wiki links)

https://en.wikipedia.org/wiki/Pratt_%26_Whitney_PW1000G

https://en.wikipedia.org/wiki/Geared_turbofan