Everything in every ECS system is done with handles, but the parent comment is correct that many games use hairballs of pointers all over the place (and they are handles with ECS). There is never a borrow checker issue with handles since they divorce the concept of a pointer from the concept of ownership.
The main feature of rust isn't borrow checking, the feature is safe low level programming. Borrow checking is Rust's default way of doing this, but it's not the only one (the standard library has Reference Counting primitives built in).
Borrow checking is a way to make manual memory management safe, ECS works around manual memory management entirely, so it doesn't need borrow checking to be safe (which is also why it's popular in C++, which doesn't have a borrow checker) but because it's Rust, you have strong guarantees about the safety of it, while in C++ you can still shoot yourself in the foot if you don't use ECS the right way.
I would argue that borrow checking is the only feature that is actually unique (more or less) to Rust. If you don't need it, you can get the rest from elsewhere, often with better ergonomics.
If you want to squeeze the maximum performance of your hardware (which is indeed the case for game engines) then all your options are Rust, C and C++. In which case is by far the more ergonomic choice on every aspects.
You could add a lot of research/experimental and/or legacy languages as well (Modula 3, Odin, D, why not FORTRAN, even) but none of them are credible alternatives to C or C++ like Rust has managed to become. And none of them are polished enough (be it in terms of errors messages, libraries, docs, onboarding material, build toolchain) to be considered superior to Rust in terms of developer ergonomics even if you set aside safety.
And again, borrow checking isn't a feature, it's an implementation of the feature which is memory safety without performance tradeoff. Even when using a paradigm that doesn't interact with the borrow checker, you're still using the feature itself (especially because all of the underlying building blocks which aren't using ECS directly are benefiting from the borrow checker's validation).
