Edge vs Level Triggered Components
As long as you stay in one clock domain, this is either completely case-dependent, or up to your own taste. There will be cases where one is is better, but no general rule of thumb. In many others, it just won't matter. What you should avoid, though, is mixing them for no reason.

The picture changes when you cross clock domains: then you should try to use edge-triggered starts. Either with handshaking, if you require backpressure, or just feedforward. Toggle signals are easier to synchronize to other clock domains.

That said, I'm not a fan of only using the rising edge on an edge-triggered start signal. I generally use both edges, either one inducing a start, which allows me to just put a few registers (and a few synchronizers) and compare the last two. It makes both the possible launching and interpreting logic easier. Of course this is only possible if both the launching and receiving blocks are reset at the same times.