FWIW I do every construction method under the sun bar wire wrapping.
It depends on the situation. For almost every MCU project that has a DIL part I'll use a breadboard or perfboard, and if there's some nastiness like SMPS or high speed mixed signal then it's either Manhattan/dead bug or I design a board. Sometimes it's a hybrid with the MCU still on breadboard. For larger prototypes with large SMD devices, I tend to use Schmartboards or other breakout boards together with the odd mini custom board as appropriate. Schmartboards are pretty good because they have ground planes making decoupling and ground connections easy with 0603 caps and 0R resistors.
For HF and below with SMD parts, it's either perfboard or I make up a board. VHF and above I always make up a board, too much time wasted trying to make any other way work IMHO.
I realise that there's a tendency to complain about breadboards, but I rarely have a problem with them and I do use them pretty much every day: I can have a basic initial rubber stamp DIL PIC project up and debugging in 10 minutes, but I am geared up for exactly that scenario as it happens so frequently. The trick is knowing when to move off the breadboard... and I'd say a 20mA charge pump really oughta work on a breadboard, so I don't know what to make of what happened. Indeed I made a charge pump last week from an oscillating PIC pin to aid bootstrapping a mosfet gate, no problems at all, although it was only a few hundred uA rather than 20mA.
Irrespective, it's time to get out that current limiting PSU of yours!