I'm watching a terrible upstairs/downstairs historical drama (I have covid, give me a break) and the fancy people are always airing their metaphorical dirty laundry or talking shit about all the other fancy people while the servants just stand there. The servants may have their own dramas but what they don't do is go across the street to Mrs. Fancy's house to tell their friends who work for Mrs. Fancy that Mrs. Snooty called Mrs. Fancy a ho and said her dress looked cheap and oh yeah Mr. Snooty pees in the parlour palm when he comes home drunk and he has a big gambling debt and a sidepiece and he and Mrs. Snooty have been fighting about it. Or whatever.
Like I get that if you got caught gossiping about your employer you'd get fired and not have a reference which could easily lead to a life of desperate poverty, but wouldn't Mrs. Snooty still have some reservations about the risk? Or just be embarrassed to be crying about Mr. Snooty's sidepiece in front of the butler? Did employers just see servants as ambulatory furniture while simultaneously servants had some insane sense of loyalty to their employers? Because that sounds crazy.
Are there any historical examples of servants causing trouble (or even preventing trouble) by not keeping their employer's secrets?
As fun as it is to watch shows like Downton Abbey, their depictions of the relationship between staff and family are notoriously awful.
As you correctly say, servants in those days were largely treated as a part of the furniture and would've been ignored by the family unless they wanted something. Most people wouldn't have reservations about gossiping in front of the servants (unless it directly concerned the family) because if it ever got out that the servant was feeding that information to someone outside the house, they could be dismissed without a 'character' (i.e a reference) which made it practically impossible to get another job in service, often they weren't able to get another job at all which lead to poverty and 'unsavoury' work such as prostitution. As one servant in 1892 wrote to Western Mail: "...no servant can expect a situation who had not got a good reference and a long one from where she last lived."
The example you give of the lady of the house crying over her husband's mistress in front of a servant is an interesting one, because it seems absurd to us today to air our grievances so publicly. But the truth is that most women simply wouldn't have cared. If a servant overheard and gossiped about it in the servant's hall, there was a good chance that her lady's maid might mention it and get the other servant punished in some way. It just wasn't worth the risk.
A lot of servants did have a strong sense of loyalty to the family they served which is a difficult mindset to grasp, but the lower classes had had it drilled into them since birth that the wealthier classes—especially those with titles—deserved their success and were somehow inherently better than them. Many, of course, rightly believed that to be total nonsense, but that didn't stop some from wholeheartedly buying into the idea.
You might be interested in reading Cassell's Household Guide. It's mostly practical tips to help a servant with their various taks, but there are a few more subtle lines concerning behaviour. In the section advising a lady's maid, for instance, the author writes: "No teaching will convey the delicate tact which proceeds from a pure mind, and the high sense of integrity which should characterise the slightest action where the interests and feelings of an employer are concerned."
As with anything, there are plenty of exceptions. Some people were more aware and would send servants out before discussing anything particularly delicate. But for the most part—especially for those who came from bigger houses and wealthier families—it wasn't something that people paid a lot of attention to. The family knew that they had all the power in this situation, and should a servant step out of line, they might do all manner of things as retribution—there are cases where a servant was dismissed without a character and was also not allowed to remove their belongings from the house; whether these were books, trinkets, or even clothes.