It's not about reacting and signing documents. An army of lawyers is useful because it allows more stuff to be done simultaneously. More people can do research on potentially relevant case law, more lawyers can write multiple different motions. This forces opposing counsel to have as many or more lawyers to react to all this.

An army of lawyers also means you can have teams all dedicated to specific tasks. Disney has lawyers that likely have decades of experience in defending against personal injury suits in their parks. That makes it far harder for anyone to succeed without a very good case.

At the scale of Disney, they likely have dedicated legal teams to many things, and even subsections of those. Take the film division for example, there are likely legal teams for copyright, contracts for services, contracts for advertising, general labor law, acting/directing contracts, location acquisition, property rental/purchase, and distribution. The same goes for the several other divisions.