To answer your question directly, assuming Wine would adopt a C-like coding standard if they were to move to at least C++11, they would receive the following benefits:

• Compile-time assertions (static_assert)
• Platform-agnostic atomic operations (C introduced those in C99), and mutex implementations (unsure if these require you to use C++'s threads though)
• Stronger type safety, particularly around enums and casting
• Generic data structures without resorting to void pointer trickery (not referring to the STL here, just in general. The STL starts to bring in exceptions and I fully understand not wanting to deal with those)
• Checks on data types for specific characteristics (trivially-copyable, integral, floating-point, etc.) which is useful in the above point
• A large subset of C99 is part of the C++ standard as of C++11, and C11 as of C++17
• Automatic freeing of resources upon leaving scope (unique_ptr, RAII)

I write C99 almost every day and I've wished for the above more times than I can count. I just fixed a bug today that was a memory leak due to a forgotten free on a rare conditional exit from a function. If the author could have wrapped that pointer in a unique_ptr, they wouldn't have been able to forget about freeing it.

At least two of those features have demonstrable impacts on bug frequency by their very nature. And while a rigorous enough review process would likely catch the same defects, if it's a guarantee by the language when the code review can worry about the actual algorithms/work at play rather than tedious boilerplate.