As I already stated, a row of pin headers are just as stable, if not more.
If you're talking the machined round pin header.... that's cool. But square pin headers are cheap as chips, even in the extra long versions, and they take wire wrap, as well. The maximum dimension of 0.035" is never going to come into play as long as you don't use single pins, ever.
But sometimes you want a test point somewhere between two components in an existing circuit, and you do not have the space to do what you suggest.
I always have the space. You can fit 5-6 wire wrap connections onto a single extra long header pin. And I got them poking up everywhere. If I make a breadboard adaptor, it's got extra long header pins poking up.
If I make a pin header cable for prototyping, it's with long header pins sticking out for probing and/or adding more connections.
If I still need another connection, it's just a matter of pulling out a component, soldering a wire to it, and putting it back in. And wrapping the other end to a header pin.
Rows of header pin test points are big part of my breadboard setup. I will set up the header pins and put some label tape down with the names of each signal. Then however else I build the circuit, I will route the signal to the header pin.
If you buy another breadboard, stick with 3M or Global. Or Wish(er?). They will handle square header pins and still work with DIP IC legs, no problem.
Sometimes when working with PCB's, I use the breadboard JUST as a place to stick header pins. I'll hot snot the breadboard and the pcb's to a backerboard. Or sometimes hotsnot pcb directly to the top of the breadboard.
When I was busy hacking a Hakko 888 board, I had to get signals from both sides and to be able to see the display. What I did to prevent a bunch of flip and flop and broken wires evertime I want to get another signal is to glue a female header on the edge of the board, and route all the signals I wanted to see/alter to the header. And then I plug this into extra long male header pin on the breadboard, which provide plenty of space and stable base to wire wrap additional circuitry and to clip the scope probes. All nicely labeled. I should have took a pic. Was pretty slick. If I want to get an additional signal, I just unplug it and take it to the microscope. No wires, no scope probes to unclip and reclip. All the junk stays with the breadboard.