The comment you are responding to is not wrong about higher async overhead, but it is wrong at everything else either out of lack of experience with the language or out of being confused about what it is that Task<T> and ValueTask<T> solve.
All asynchronous methods (as in, the ones that have async keyword prefixed to them) are turned into state machines, where to live across await, the method's variables that persist across it need to be lifted to a state machine struct, which is then often (but not always) needs to be boxed aka heap allocated. All this makes the cost of what would have otherwise been just a couple of method calls way more significant - single await like this can cost 50ns vs 2ns spent on calling methods.
There is also a matter of heap allocations for state machine boxes - C# is generally good when it comes to avoiding them for (value)tasks that complete synchronously and for hot async paths that complete asynchronously through pooling them, but badly written code can incur unwanted overhead by spamming async methods with await points where it could have been just forwarding a task instead. Years of bad practices arisen from low skill enterprise dev fields do not help this either, with only the switch to OSS and more recent culture shift aided by better out of box analyzers somewhat turning the tide.
This, however, does not stop C#'s task system from being extremely useful for achieving lowest ceremony concurrency across all programming languages (yes, it is less effort than whatever Go or Elixir zealots would have you believe) where you can interleave, compose and aggregate task-returning methods to trivially parallelize/fork/join parts of existing logic leading to massive code productivity improvement. Want to fire off request and do something else? Call .GetStringAsync but don't await it and go back to it later with await when you do need the result - the request will be likely done by then. Instant parallelism.
With that said, Rust's approach to futures and async is a bit different, where-as C#'s each async method is its own task, in Rust the entire call graph is a single task with many nested futures where the size of the sum of all stack frames is known statically hence you can't perform recursive calls within async there - you can only create a new (usually heap-allocated) which gives you what effectively looks a linked list of task nodes as there is no infinite recursion in calculating their sizes. This generally has lower overhead and works extremely well even in no-std no-alloc scenarios where cooperative multi-tasking is realized through a single bare metal executor, which is a massive user experience upgrade in embedded land. .NET OTOH is working on its own project to massively reduce async overhead but once the finished experiment sees integration in dotnet/runtime itself, you can expect more posts on this orange site about it.