Why are visual approaches (in the US) preferred by US pilots, but not European pilots?
Capacity
If a visual approach is offered, and it gets accepted by the pilot, the airport can declare higher capacity. Whereas IFR procedures reduce the airport's capacity.
US and European airports handle their slots differently. Europe plans for the worst, which can on good days limit the full potential. The US plans for the best, which can on bad days end up in huge delays.[1] I've also explained it here. In short, if in the US they started offering more IFR procedures during good weather (VMC), they won't be able to back up the declared capacity.
Generally, airports are closer to the cities in Europe than in the US (apart from coastal cities) due to the different city planning (old vs. new). Airports in close proximity to cities have approach procedures that are designed for noise abatement, and are best flown as precisely as published.
Familiarity
So in general, the pilots are used to different systems. Also pilots that don't fly a lot in the US—like foreign carriers—are not as familiar with the surroundings as local pilots are. A Flight Safety Foundation briefing[2] discusses the risks associated with visual approaches for airplanes heavier than 12,500 lb:

Accepting an air traffic control (ATC) clearance for a visual approach or requesting a visual approach should be balanced carefully against the following:

(...)
Crew experience with airport and airport environment.


It's about knowing what to look for, especially when a particular visual approach does not have a visual approach chart.


An example of a visual approach chart showing landmarks; airnav.com

Use of ILS in emergnecy
Regarding the specific video about the bomb threat and the pilot's request to have the ILS turned on, for that scenario in particular it's because an ILS approach is less demanding,[3] especially in such high workload environment. (Also the flight was diverting mid-route.)

Poor professional judgment/airmanship (...) Decision to execute a nonprecision approach, instead of an ILS approach, in demanding conditions to
expedite arrival (...)[3]


1: Odoni, A., and T. Morisset. &quot;Performance comparisons between US and European airports.&quot; Proceedings of the 12th world conference on transport research (WCTR), July 11–15. 2010. (paper; presentation)
2: Flight Safety Foundation ALAR Toolkit. ALAR Briefing Note 7.4 — Visual Approach. (PDF)
3: Khatwa, Ratan, and Robert L. Helmreich. &quot;Analysis of critical factors during approach and landing in accidents and normal flight.&quot; Flight Safety Digest 17.11-12 (1999): 1-2. https://www.jstor.org/stable/44729511