The question states
For a large website developed in PHP, is it necessary to have a framework?
And the answer is yes.
The very existence of frameworks changed the development scene. Frameworks let you do mundane task fairly quickly, which leads to more time demanding clients and project managers. Not to mention code standardization and maintainability, among others.
The non framework way
Let's say for a moment that you need to develop a very, very large website using raw PHP. The first thing you would need to do is to define your project's directory structure. Next you start thinking about what features you would need: a database library, some kind of "utils" library, etc.
Many of these requirements will come on the fly. You'll start changing and moving stuff constantly until you're satisfied. How do we map URLs to controllers?, how do we map the models to the database? are some of the questions that will arise and will consume valuable development time.
You start coding and stuff starts to get complicated. In order to standardize your code (so you and your team don't collide with each other's work) you start creating your own inside rules. Unit testing comes into play and you realize your classes are a pain to test, so you start to refactor everything, once again.
By the time, 2 months have passed, you just made a framework from scratch and your project is far from finished. Stakeholders start to get anxious and the project manager start to get nervous. You get the idea.
The framework way
Rules are already defined. All you need to do is learn how to use this new framework. In a matter of 1 - 2 weeks (based on the time it took me to learn Django and Laravel) you're already able to tackle down your project. Everything's fluent, you don't need to worry about trivialities like URLs and ORMs.
Most of the frameworks are open source, they're endorsed by the community, so you don't need to worry about trivial (and often overlooked) security problems like XSS and CSFR. You're also ensuring your application will always be up-to-date with community standards and new practices.
Using a framework doesn't ensures maintainability, it promotes it. Unit testing, community defined rules, standards and structures are always a plus. If by any chance you need a new developer just ask HR for a senior or guru on X framework, that way you skip the overhead of explaining your inside monster of "on the fly" framework and just dig right in into your business logic.
Large projects mean stakeholders and investors and that means time and money. They don't care about the application's performance or the milliseconds you save by not using a framework. All they care is results, results and money of course.
By using a framework you open a huge set of professional opportunities for yourself. What if your local "Google HQ" is looking for a senior Laravel developer? You suddenly realize how archaic your self-made framework is and how it can't compete with an "edgy" community driven development. Even worst, you spent a year working with a framework no one else except you or your company will use and you have no other option but to feel outdated.
That's all I can think of now.