I know you asked about “safest” not simply “safe”, but let me make some point.
There is no such thing as safe method, because any safe etchant would also be useless. Chemical etching of copper works by turning copper into ions, which then dissolve into water. That happens by copper losing electrons to the etching solution. Which means: it must be an
oxidizer. If you don’t want to wait a month for your PCB to be made, it must also be reasonably strong one. And those are not good for you, no matter which you choose.
This leads to an important conclusion. Some substances will be a bit less risky to use, some a bit more. Some may produce dangerous fumes or heat. Some may be cumbersome to store. But you
must observe basic safety, because you are dealing with something that is doomed to pose a risk to do its job.
When you accept this, your original question gains a new perspective. If you wear nitrile gloves while working with your etchant, suddenly it becomes of tiny importance if handling the PCB causes irritation to your skin. It shifts towards skipping options that pose risks you can’t easily avoid: like producing fumes, requiring strong acids or giving spontaneous combustion.
The second important thing is: while you may get relatively safe etchants, the waste product of the process is always nasty. That’s because it contains
copper compounds dissolved in it. And copper is toxic. While the amount hobbyists produce is not enough to care about disposal safety, chronic exposure is something you want to avoid. So no eating while etching, cleaning surface you worked on (if that’s your desk or kitchen table), washing hands, no re-use of etching container and tools for food, good rinsing sink with water and cleaning it.
As for recommendations, I only used sodium persulfate. It is widely used here by hobbyist, available as “etchant B-327”. As already noted above it requires elevated temperature to work well, but do not panic: for simpler PCBs you get away without using a heater: hot water (
below 70°C) will stay warm enough sufficiently long. Moving the PCB with an old plastic ballpen case. May require a second batch. Proper gear is needed for complex or small pitch boards. Storing it requires reasonable protection against moisture ingress and contact with anything that could react with it. It comes in plastic bags. Keeping that bag sealed and away from pointy items that could puncture the bag is enough.