Oh, this again. For most of my "professional" career, I and just about everyone I know really only owned one instrument. My keyboard player didn't bring a "backup" DX7 and my guitar player might have brought an acoustic and an electric but I don't ever recall seeing two electrics on the stage.
In that decade I probably played 3-set shows anywhere from 3 to 5 nights a week, depending on the time of year. Summer was busier. Count up all those shows over ten-plus years and in all that playing time I broke two strings. Both times I was able to get by with what I had; I think once I just tuned up the remaining strings and played that way until the end of the set and the other time we were close to the end of the set anyway so we just finished the song and took our break a little early.
Given the number of failures I experienced over the many hours of stage time, I think that the likelihood of failure is overstated. Given the type of shows I tend to do nowadays, I feel like I'm taking myself a little too seriously if I'm dragging along another instrument that I'll likely never need. If I have to bring two then I'm going to play both of them, so a four and a five or something like that. But most of my shows are not shows where I'm going to be unable to accommodate a problem if it happens.
I do take backups of the most likely failure points. Extra cables, strings, batteries, tuners, power cables, all the things that typically do or can fail. I don't care if someone else feels like they need a backup instrument, but it seems like overkill for the situations I'm usually in.
The "likelihood" of failure is low; the consequences can be great depending on the gig. I have insurance on my car and house even though the likelihood of their destruction is low. And I have never totalled my car or burnt the house down. Still glad I had the insurance all those years.
Also, almost every serious guitar player I know and play with brings 2 guitars. poopie happens. If it's a local gig close to my house and not a big deal, I have gone without a backup. To my detriment. Every time I travel, and every time its an important gig regardless of location, I have a backup. And almost always do bring one even for casual gigs - I don't like the stress of having to come up with a fix on the fly, regardless of who I am playing for, or where. And yes, I have had on-gig failures. You play enough gigs, it's going to happen. On one of these instances I was lax on my rule, and didn't have one... 75 miles from home...found out I had a problem 30 minutes before start time, somehow managed to locate and obtain a substitute bass. Nerve-racking! After that... pretty much ALWAYS. Don't let a lack of problems before lull you into a false sense of security.
One of my most harrowing experiences, which convinced me to ALWAYS bring a back-up for any gig of consequence, was one where I could not bring a back-up - a tour in Europe where I could only bring one (air travel). On stage in Belgium, in front of thousands, the first song starts with about 4 measures of drums before the full band comes in - my theretofore failure-free bass, even in sound-check - no sound!! Loose connection in the vol pot. I wiggled it, jiggled it, turned it up and down- nothing - then with only a split second to spare, fear shooting through me like electricity, I gave it a desperate turn, and it came back. Literally, a quarter of a second before I needed to come in.
I cannot imagine what would have happened if I couldn't get it back, being abroad with no back-up. I did not touch the volume control the rest of the show, worried constantly that it might just go out again on its own. Got it fixed the next day but jeez - nerve-racking.
A back-up for everything = peace of mind. IMO, professionalism means never being an obstacle to a flawless performance. People will not hire someone who is unprepared for the unexpected, in a professional setting.
Final note: I saw Marcus Miller's bass fail on stage once. He had a backup and so he was up and running again in 20 seconds. If it can happen to a guy with a professional sound tech to prepare his gear, it can happen to me and you.