Yes, it will make refresh slower, but searches faster. I think the near realtime delay would be at least the same with eager global ordinals but likely to be worse compared to lazy global ordinals. (I think whether the cost is spend at a refresh or a search request is irrelevant to most of the users). Unless you completely denormalize your documents a 1 second near realtime delay isn't achievable with parent/child at scale.
Eager global ordinals is a form of warming, which is part of the refresh being executed. A shard (Lucene index) consists out of segments, changes are written into new segments and existing segments maybe merged into a new larger segment (previous segments are eventually removed). Changes (the new segments) aren't visible immediately via the search api, the refresh updates the view (current visible segments) the search api is using periodically on each shard independently and eager global ordinals loading is being triggered on this level.
In my opinion eager global ordinals loading is better than lazy global ordinals loading in the case where search requests with
has_parent queries are executed frequently. If a Lucene index changes each second then when multiple search requests are executed in parallel each search request is likely to have its own view of the Lucene index and this causes global ordinals to load more often then once per refresh, which can increase the load significantly on the ES side. If searches are less frequent then it makes sense to keep the default (lazy global ordinals loading).
Also you can consider (if your data allows this) to split your data up in more indices based on certain properties in your data (for example user) or use routing (both at index and search time) with your current index strategy. With the main goal of making it less likely that changes effect all shards / indices.
Also adding a few of more shards in general also helps as the global ordinals on a shard level are smaller and can be executed more concurrently. However don't create too many shards as that can result in other problems. If more than a few shards need to be created then also add more nodes.