But this is a hand-wave.
This is a situation where both vehicles got explicit permission from someone who's supposed to know what they're doing. These sorts of runway crossings aren't unusual - and this one was responding to an emergency - and at a place like LGA there's always gonna be a plane on approach.
The difference between "hold short at runway 22" and being on runway 22 is much less than 30 seconds in some cases.
It's obviously the right choice to give the PIC the information via avionics in a graphically concise way that highlights this potential runway contention because it is real and pilots are expected to adjust their speed to maintain the right sequence.
When it isn't possible, which does happen, IE a plane ahead is slow to clear the runway or to takeoff, pilots are expected+required to execute a go-around.
> The difference between "hold short at runway 22" and being on runway 22 is much less than 30 seconds in some cases.
What is the typical minimum temporal separation? I would have expected at least 45 or 60 seconds given the cost of a plane and the imminent threat to life.
https://www.nbcnews.com/news/us-news/investigators-search-an...
> Jennifer Homendy, chairwoman of the National Transportation Safety Board, said at a news conference Tuesday that the airport uses a safety system called ASDE-X to track surface movements of aircraft and vehicles.
> "ASDE-X did not generate an alert due to the close proximity of vehicles merging and unmerging near the runway, resulting in the inability to create a track of high confidence,” Homendy read from an analysis of the system’s performance.
* https://www.radiantvisionsystems.com/blog/worlds-most-advanc...
Not sure if you can do something on a 'simple' HUD that many planes have, so you could see objects in your flight path.