No browser I'm aware of implements that under normal circumstances.
If you run with the debugger open they do tend to insert a lot of instrumentation to enable pausing/breakpoints/etc but that comes at a massive performance hit - in some cases 'debugger open' WASM is slower than JS.
My guess is that the performance overhead of inserting it into wasm was judged unacceptable. It's a little odd to me, I was surprised when I discovered I couldn't realistically pause a tab running wasm benchmarks - often when I try to close the chrome tab it just hangs.
The runtime I work on (.NET) inserts safepoints at back branches in some cases specifically to enable pausing for GC. It wouldn't surprise me if using WASM GC would cause the runtime to start inserting safepoints as well.
(I presume the browsers implement something similar, but I've not tested it to see.)