Saying that the work could not have been done without it is basically admitting that you got papers outside the subscriptions of your institution. It is also against the policy of all the institutions I know, so could be grounds for termination. And again, this could apply to your co-authors as well.
I am really not endorsing the subscription model, but putting something like that in your articles' acknowledgments really is not smart.
I will allow that there may be some harms which can result from unlimited copying of anything you want— but they bear no resemblance to the collision of automobiles in the midst of a busy intersection.