Breakout boxes still appear here & there. These days, though, they're pretty much no longer just passive, but have internal signal conditioning so the measurement devices don't load the line. In fact, they often times redrive the lines.
I own a couple of RS232 Breakout Boxes exactly like the one above, and have made myself a couple over the years as well (mostly dedicated to 9-pin). I once used to look after terminal controllers with 8 RS232 interfaces, and built some special test gear for them, that story is here (and above, How to have fun with Klunky old terminals) if you have not read it before.
I still have a fair bit of serial-interfaced gear to test, and use breakout boxes from time to time. Younger colleagues are often amazed when I hook one up to an obstinately dead bit of equipment and say "Well, your RTS line is dead!"
I must admit that USB is fairly reliable and has removed the need for fiddling around with wires to get the right cable configuration, but sometimes it is difficult to know what is wrong if it does not work.
For buses like I2C and SPI you can use oscillloscopes to troubleshoot now, I guess that would be the nearest equivalent. It's not so much signal integrity that has made these devices obsolete, it's the speed of the interfaces we use now, and the fact that their physical simplicity (small number of wires) means that most problems will be at the protocol or software level.
I improvised a USB breakout box from several USB cables that I had and didn't need any more. I soldered the wires to a small piece of PC board, with bare wires going across a small gap (and they can be opened, if needed). It allows me to check basic DC levels and presence of signals--but nothing else. But, hey, many times it is these basics which are causing problems, so it's a quick-start to check and verify.