The lab PSU is needed, that you dont die, the circuit doesnt die, and it makes your life easier. It has current limit, which can be controlled by a knob, so if the circuit misbehaves, it will not start a fire or blow up. If you are serious about electronics, this should be the second thing you buy, first being a multimeter.
BTW, there is nothing wrong with the power supply you showed. When I was young, I've used a power supply designed for model train speed control. Thinking back, I should have bought something nice, it would be still usable now.
There are OK quality chinese power supplies, the trick is to know which one. Farnell sells Tenma branded power supplies, they are OK, has some quirks, but they work, and they are safe. The cheapest is about 70 EUR.
mod: Now I'm seeing, that yours is also Tenma.