It seems like there are two parts to this question, and, perhaps, OP will tighten up the language a bit.
Do we need to use detergent in washing machines?
Depends on what you want when you wash your clothing, I suppose. It's possible, that for some people, the amount of dirt and soil removal from just rinsing is adequate. That's a matter of personal preference.
But if you are asking -
Does detergent make any difference vs plain water?
That's a different question. Generally speaking, yes, there is a difference. But what about Ruth Goodman's claim? It could still be true, and the answer that detergents do/can make a significant difference could also be true. How?
NOT ALL WASHING CONDITIONS ARE THE SAME
We really don't know what conditions Ms Goodman used for her washing all of those years. Detergent, water conditions and level soil being equal, some machines do a better job of washing than others.
A look at performance
We've reviewed dozens of washing machines. While we still have a long way to go before we've tested all of the models on the market, top-load washers have earned our best (and worst) stain removal performance scores so far.
In contrast, the front-load washers we've reviewed tend to fall somewhere in the middle performance-wise, with a few outliers on either side.
CNET Washing Machine Buying Guide
Equal machines and soil level, having hard water will inhibit the ability to clean clothes.
Soap and detergent simply don’t work as well in hard water. Those dissolved minerals hamper the effectiveness of many cleaning products.
That’s because minerals like calcium and magnesium prevent water from mixing with detergent to form a solution. As a result, soap scum gets left behind. The same white, chalky substance in your sink and shower gets on your laundry. Sometimes, the stuff that sticks to the fabric is referred to as “detergent curd,” which sounds even more disgusting.
The result is dingy looking clothes that don’t feel completely clean after they come out of your dryer. The residue on your clothes will even attract and hold more dirt as you wear them.
WaterRight Group Blog: Is Hard Water Ruining Your Laundry?
NOT ALL DETERGENTS ARE THE SAME
All other factors being equal, some detergents to a very good job (some perform better or worse in hard-water or particular water temperature situations, as well), and other ones are poor quality, just like with any other product.
Consumer Reports (CR, for short) is known for objective product testing. They test many kinds of laundry detergent under many different conditions, and there are definite differences between the detergents. By definition, if detergents really did nothing, you wouldn't see a difference. However, as a control group for comparisons, CR would also run a plain water wash for their tests, so we don't really have to speculate about that. The worst of the detergents performed barely better than plain water. The best ones did much better.
Consumer Reports recently tested over 50 different detergents, including a new one from homemaker Martha Stewart. They set out to find which are worth your money, and which ones are like putting money down the drain.
In the study, all detergents faced the same challenge: washing in both top-loading and high-efficiency washing machines. Material was also washed using just plain water in order to see what difference, if any, the detergents made.
Martha Stewart's detergent, for example, claims it "removes tough stains." Consumer Reports used swatches riddled with some "tough stains" (rings around the collar of a shirt, wine, grass and blood, just to name a few) to see if any differences could be observed.
After washing the swatches, testers used a special device to precisely measure how much of the stain had been removed. So how did the decorating queen's detergent do?
In conventional top-loaders, there was barely a difference between the swatches washed in Stewart's detergent, and those washed in plain water. The results of her detergent in the high-efficiency machines fared better, but only slightly.
Wave3 News: Could plain water clean as well as high-priced detergents?
(Note: Consumer Reports' testing results are behind their pay wall, otherwise I'd directly cite their results and commentary for you).
So, according to objective tests, decent-quality detergents did make a significant difference in cleaning laundry vs. plain water. They do use measuring equipment to discern between results that are close together, but getting a stain completely out vs. mostly out vs the stain pretty much being there would be a significant difference to even a fairly casual observer.
CR’s testers wash fabric swatches that are saturated with blood, chocolate, red wine, dirt, grass, tea, and body oil. These are tough stains to remove, and we use them to challenge the detergents so that we can detect real differences among them. Using cool water, we wash the swatches in three identical washers with each detergent. We then allow the swatches to air-dry. (A dryer is out of the question because the heat can alter the stains.)
Testers use a colorimeter, a device that measures color intensity, to see how much of the stain remains on each dry swatch, compared with stained swatches that have been laundered using only water.
Consumer Reports: Tide Beats Persil in CR's Laundry Detergent Tests (behind paywall and not viewable without an account)
However, the worst ones didn't. And it wasn't necessarily "cheap = low-quality," as Stewart's product carried a premium price tag.
If Ms Gordon had any combination of inferior machine, poor water conditions, not following recommendations for her detergent, or inferior detergent quality, it is entirely possible that she saw no appreciable difference in her detergent vs plain water, while it also being true that detergents can make a big difference, as well.
As product manufacturers like to remind us, "Results May Vary."
Addendum: Here's an article, directly from CR about which machines and detergents people should avoid, that they posted, available for free to the public, as a service. It doesn't add a ton more detail, but I figured, straight from the horse's mouth instead of second-hand adds a bit more backing to my answer:
With laundry detergents, you can waste money a couple of ways. Detergents that don’t clean well might require you to redo a load. And if you don’t measure the detergent, you might end up using more than you need.
These detergents were no match for common stains such as body oil, dirt, and grass. You’ll find much better detergents in our laundry detergent ratings.
Among liquid detergents, Xtra ScentSations and Trader Joe’s Liquid Laundry HE detergent clean only slightly better than plain water. You can also skip Xtra Plus OxiClean and Sun Triple Clean.
Consumer Reports: Laundry Products That Waste Loads Of Money