People with weaker triceps will often switch to a wider grip to put less focus there. And, people with weaker triceps will do other side training to strengthen their tris - be it addition sets of close-grip bench, tricep isolation exercises, band bench, etc.
		
		
	 
If they are doing full range of motion bench press, then they are still doing bench press - nothing wrong with that. Switching to mixed grip or hook grip for the deadlift is no different - it is still the deadlift. Using straps, however, is an artificial aide, just like a spotter helping you finish reps of bench press.
	
		
	
	
		
		
			What using straps does is allowing you to continue working the rest of the chain while you do additional exercises on the side to strengthen your grip. That way, when your grip finally does catch up, your deadlift as a whole is stronger than if you'd just patiently waited.
		
		
	 
Well, this will certainly vary from person to person, but in my experience, 99% of people who use straps will NEVER have their grip catch up. That is, grip strength will ALWAYS be their limiting factor on the deadlift so long as they always do their heavy sets with straps. 
	
		
	
	
		
		
			For me, my grip didn't finally catch up (and far surpass) my deadlift until I started doing rock climbing. Had I not used straps, once I started climbing and strengthened my grip sufficiently, my deadlift would have been 50-100lbs behind in all likelihood.
		
		
	 
My experience was different. So long as I used straps, my raw deadlift lagged ~30lbs behind what I was doing with straps. When I finally ditched straps entirely, it took only a month or so to make up that difference using a simple plan:
* I dropped by deadlift to 40lbs less than what I could handle with straps
* My routine had me deadlifting once a week and I increased the weight 5-10lbs each week
* I did other grip heavy exercises (pull-ups, rows, shrugs) once per week in between deadlift sessions
In other words, you can treat it as a simple deadlift deload that lets you focus on technique. In fact, I ended up with a 
better deadlift at the end, so I didn't lose any progress. Of course, in the worst case, it might take 2 or even 3 months for grip to catch up. This still won't be anywhere NEAR the 50-100lbs difference you mentioned. For most beginners, I'd wager the difference will be 0 in the long run if they properly dedicate themselves to it.
Oh, and IMO, the fact that your grip suddenly caught up when you started rock climbing is evidence that you weren't doing enough (or the right type of) grip work before.