Here’s what I think the idea is. Ocean upwelling cools the surface by placing cooler water at the surface. It then emits less warmth to the atmosphere. Lessen this upwelling and the surface is warmer, emitting more warmth to the atmosphere. Increase this flow, with La Nina conditions and the atmosphere is cooler than it otherwise would be.
Global ocean temperatures to their full depth would be warmer with increased upwelling as cool water that receives sunlight will keep more of it when compared to warmer surface water. It evaporates less. Ocean temperatures with decreased upwelling would emit more warmth from evaporation so they would be cooler if everything else is equal.
A La Nina circulation would be vertical from the ocean depths and then horizontal along the equator. It would involve some of the coolest liquid water in the best place to warm it. We know the ocean depths have sustain with their massive thermal reserves.
The oceans have warmed as has the atmosphere. So I think this line of thought cannot explain why both have occurred at generally the same time. However if upwelling was high, the oceans would warm while slowing GMST rise. There is a saying, The hydrological cycle speeds up with warming. If this is the case, change may be limited or moderated.
Let’s now apply this to the glacial/interglacial cycle. A descent into a glacial would involve oceans cooling. Less upwelling, more emission to the atmosphere. The system slows. This cools the oceans. The oceans are less efficient at warming themselves. Compare this the efficient warming during a La Nina.
An ascent to an interglacial involves oceans warming. Upwelling increases efficiency by placing the cool water in the tropics to be warmed. As the oceans cooled during the descent their loss of energy actually increased future efficiency of warming.