Chris, you wrote:
"It was the equation of time that was throwing me off."
Yes... I know! :) And I should emphasize that there was nothing else wrong at all with your analysis. You just missed that one step at the end. It wasn't a mathematical error, merely an oversight in using the method.
There's a funny thing about you're sight data: it would be diabolical on an exam. For starters, your lat and lon were nearly identical. The lat was about 47°, and the lon about 48°. I looked at latitude instead of longitude at one point while examining your numbers, and it threw me off because it was close, but not quite right. This is just the sort of thing that leads to difficult-to-diagnose errors on paper. Exam designers love that sort of thing!
In addition --and relevant to your issue here-- you took this sight on a date that's very close to one of the zeroes of the equation of time. I like to teach that the equation of time is zero around some "memorable dates": Tax day (Apr 15), School's out day (Jun 12? ...that one's a stretch), Labor Day (ok, Sep 1 ...close enough), and Christmas. The two extremes are Election Day (Nov 4) and a week after Groundhog Day (Feb 10). Since you took your sight just a couple of days after one of the zeroes in the equation of time, it was hard to detect the problem. It didn't yell at you. By contrast, if you had taken your sight in early November, you would have seen a 16 minute error corresponding to four full degrees of longitude. An error that large would have been obvious trouble, and it would have made you go back and look for a problem. As it was on Sep 3, you could imagine that the small resulting was just some sort of "round off" error in an old-style computation.
"Would it be safe to say that this method of finding longitude and local apparent time was replaced with by more modern longitude by chronometer method and UTC?"
No, that would not be safe. Most un-safe. Heh. :)
First, many navigators were still doing longitude by time sight as late as the Second World War. The history of navigation is more varied than the cartoon histories in the navigation manuals (Bowditch, e.g.) would imply. Next, the expression "longitude by chronometer" was British "slang" for what we call a time sight today. So that's the same thing really. In the 19th century, these were often called sights "for the true time" by which they meant local apparent time... Sun time was "true time". As for UTC, it's only trivially different from GMT, and navigators depended on that from the late 18th century onward.
You can use the math of time sights in modern celestial navigation with very few compromises, and this methodology has a number of nice advantages. If you're worried that you don't see a line of position in this, there are two easy ways to generate one. You can use any of the standard methods, by computation or by graphical lookup, to get the body's azimuth. Plot your calculated lat/lon, then draw your LOP perpendicular to the azimuth passing through that single point. No problem. Instead of calculating the azimuth, you can just run the time sight math a second time with a slightly different input latitude. That gives you two lat/lon pairs, and you draw your LOP through those two points.
The celestial line of position is the key navigation construct that we get from each celestial sight. We can get it by the intercept method or by time sight calculations. It doesn't matter which we use. Historically, the intercept method offered slight reductions in math work, and that was important when computation was expensive and time-consuming. Those days are gone.