It's worth noting first that the "vintage" craze and inflated prices on 1950's-60's gear is both a recent invention of the last 20 years or so and not at all representative of the average electric guitar that teenagers with part time jobs are buying.

Minimum wage in the US, adjusting for inflation, was roughly 15% higher between 1950 and 1970 than it is today, so focusing on minimum wage workers, our predecessors had a slight advantage in real wages.

A standard Gibson Les Paul in 1960 cost $247, or about $2,400 today. A Gibson Les Paul standard today costs... almost exactly the same.

On a budget, however, an inexpensive beater guitar from the Sears catalog would cost our aspiring rock star $39, or about $380 today. Today, a comparably low end beginner guitar runs about $150. "Comparable" is being used loosely here though; the average beginner instrument today made with automated CNC is a vastly better built, more playable instrument than the cheapest guitars assembled 50-60 years ago when the electric guitar was in its infancy. In spite of slightly lower wages, our modern day guitarist would be much, much better equipped for the same amount of work hours.

Conclusions; electric guitars of an ultra premium vintage are more expensive today as the category didn't exist until at least the 1980s. Ordinary high end guitars have remained roughly the same, adjusted for inflation; whether the quality is better or worse is hotly debated. Low end guitars have become significantly more affordable and higher quality even in light of slightly lower income at minimum wage.

TL;DR: certainly less affordable than it would be today, but a basic electric guitar and basic amp could be had for about 40 hours of minimum-wage work, each.

There are a lot of layers to your question, and I'm going to try to address each of them. First, although you did see a lot of young musicians playing premium guitars like a Les Paul through premium gear like Marshall amps, those products were as much luxury items then as they are now. Artists were able to get these instruments thanks to the support of record companies, who would typically advance cash for an artist to support recording and touring activities.

For reference, a Fender Stratocaster with a tremolo and a custom color cost about $290 in 1960. That's about $2800 in today's dollars. A brand new American made Stratocaster costs $2499. A Les Paul Standard set you back $235 ($2320 in today's dollars) in 1960, while the cheap model would be $110 ($1086.) Those same guitars cost $2800 and $1700, respectively, today. All things considered, that's pretty remarkable price stability, but still outside the reach of most teens.

Most musicians working on their own budget were more likely to get a less expensive option. In 1960, this would likely be sourced out of the Sears Catalog. Sears had a house brand, Silvertone, which counts the likes of Bob Dylan, Buck Owens, John Fogerty, George Harrison, Elvis, and Jimi Hendrix among their famous users. "Silvertone" guitars and amps were made for Sears by a variety of manufacturers including Danelectro. They were cheap, well-made, and sounded pretty good. Some of them were also famously made out of masonite and tape, and although this was the cheapest possible construction method, it made for a quirky guitar whose idiosyncrasies are still sought-after today.

Minimum wage in 1960 was a convenient $1/hour. This translates to about $9.60/hour today, but I'll get to that in a second. A basic Silvertone electric guitar could be had for $37.95. You could get an amp with an 8-Inch speaker for the same price, although if you wanted an amp that you could play an actual show with, you'd likely need to upgrade to the 12-Inch model, which would set you back $68. So you're looking at ~110 hours of work to be able to afford a guitar and amp that will let you play a gig. I think this would be reasonably affordable for lots of teens, especially if mommy and daddy are helping.

You mentioned the current expense of vintage and vintage-like equipment, and there are some important points to be made. The biggest is that technology like vacuum tubes, while esoteric and expensive today, were being mass-produced in 1960. Similar to how today's guitar amps use chip amplifiers that aren't all that different to what you'll find in a home or car speaker system, the vacuum tubes used in 1960 were readily available, and their use in consumer electronics meant that they were cheap.

Tubes are also really cool musically. Tube amps respond differently depending on what is being played through them, and this unpredictability gives them a wonderful lifelike quality that is hard to replicate with digital amps. Even though digital amps have progressed to being technically superior, the musical nature of tube amps keeps them in high demand.

The "vintage" and "american-made" things have a lot to do with corporate shakeups in the late 60s and early 70s. You mentioned the Les Paul and I made a point about a Stratocaster. In 1960, the companies that made these instruments, Gibson and Fender, were American companies with American manufacturing that were competing on the prestige of the quality of their product. In 1965, Fender was sold to CBS and in 1969, Gibson was sold to a South American brewing company. Both companies' products suffered significant loss of quality in the following years, which led to huge demand for guitars that were made before these corporate acquisitions. Fender famously moved manufacturing overseas, and the difference between American-made, pre-CBS guitars and their Japanese-made counterparts was stark.

Interestingly, Fender's Japanese manufacturing operations eventually started cranking out some really good guitars, and now there's specific demand for Fenders that are made in Japan. Even so, the sense that the guitars were better before the company got sold has led to a pervasive belief that the old ways were better, which helps drive demand for products that are made to the same spec as the pre-sale instruments.

All that said, entry level musical equipment is far cheaper today. The Silverstone guitar that I linked above is a "single pickup guitar with tone and volume controls." Its $37.95 1960 price tag would translate to $375 today. Here's an upgrade that you can buy for $175. Even with the current federal minimum wage of $7.25/hour (compared to $9.60 if it were equivalent to 1960's $1/hour,) the Squier would only take 24 hours of work to buy. You can get a decent amp for about $250 that is far more powerful and versatile than the amps available to a teenager in the 60s.

Source: Mostly linked, as well as many years in the Music Instruments industry.