They will effect everything else in more ways then just your other exercises.
The name of the game is progression.
To keep making gains, and to keep making progression one of the best ways to do that is by simply adding weight to the bar. Now this cant always be done, nor done on a consistent basis, so other forms of progression start to show up there...
Straps are only going to take you so far, and then when you reach the end of the road with straps helping, what are you going to do? your grip will be horrific due to the fact you used straps to begin with.
What your doing is essentially relying on a crutch, thats all they are. Other things like belts/knee wraps are a little different and should be used with caution and intelligence. Straps are doing absolutely nothing for you, when if you would just take a few weeks to lower the weight, work on your grip and let it catch up...you will be relying on your natural grip strength and it will improve and allow you to keep progressing with heavier weight.
Stingray presents a great example of what straps can do to you.
Do you really want to rely on a crutch like that and force yourself to use straps on everything else including shrugs/pull ups/rows just because your grip just cant handle it?
JMBS
You mean to tell me powerlifters don't use straps? I understand that using them doesn't allow you to develop the grip. I try to only use them on heaviest sets. They might mess with wrist mechanics too and put strain where it's not good. Other than that, what good is a super strong grip going to do for me? And is it something easily developed or is it going to hold me back on chins, deads, rows, shrugs?
6'3" @213
Squat 1x20x275
press:200, Deadlift:475, Bench:300, P.clean:235, Squat:385
"The only failure that is final is to stop trying to improve"