Is it okay to rely on CocoaPods when deploying an app?
Yes, you can add as many cocoapods to your app as required, it makes no difference regarding the count of packages.

Many of the most popular apps have a dozen or more packages.  However, whether or not you should be using that many packages is an entirely different question.

Also, your point about the MIT license is a good one.  You should also be acknowledging all these package creators within your app somewhere.  For example, in my app's Settings page I have a tableview row that lists the packages my app uses.  App Store review will not check for that but it I think it's required with the MIT license.