It seems various things have changed recently with Google judging from both old and new posts:

1) Multiple variations of search term

Previously (e.g. 4 years ago) people on here were saying that when an article about them on Google is delisted, it is only delisted for their name as a specific search term. E.g. "John Smith" but not "John Smith of Trumpton".

That seems to have changed? Whereas previously I was always on page 1 on Google, under most things with my name, I got my self delisted last week, just giving my first and last name as a reference, and now I can google:

John Smith Edinburgh
John Smith Info
John Smith 28 [my age cited in article]
John Smith [quote words used by judge]
Etc etc,

All clear! I even switch to the "News" section at the bar at the top, and type "John Smith 28 Edinburgh court judge", and I get an obscure fluffy nice story about me (pre-conviction), but not the criminal episode.

In fact, in order to find the news article again, I have to omit my name entirely and type a unique phrase from the article.

2) USA listings of old tabloid stories

Using Issearchfrom and selecting North America, I have to work very hard to discover UK citizens whose reputations are comprehensively trashed in the UK Google. For some people only 1 in 10 articles remain. Whereas previously I was on page 1 of USA also (unique name, tabloid story)

3) Cooperation

It seems Google defaults more to ushering through takedown requests, rather than refusing them?

I've noticed Number #1, but might just be confirmation bias on my part. 

I still do periodic checks and will usually search like:

1. John Doe
2. John Doe [CITY]
3. John Doe [CITY] sex
4. John Doe [CITY sex offender

I type "7463782" into Google, the identity of the article within the URL, and it comes up as the only hit.

I type "Smith 7463782", no hits!

Incredible.

(name and number changed to protect the guilty)

If you know where to look you can still find me, or at least a couple of articles. The main stuff has thankfully disappeared after Facebook banned a vigilante account.

I've come to terms with the fact people can still find out my past if they're invested enough but by the same token if they're digging that deep into me I'd suspect we'd be close enough that I likely would've told them anyway.

Is there any advice on here about having newspaper sites take it down on the same grounds, or does that incur a Streisand effect?

I think you can certainly reach out to the newspaper(s) in question and I don't think it'd cause a problem, however, I will be shocked if their answer is anything other than "No". Media integrity, archival purposes yada yada, you know. The only way I could see a newspaper taking down articles is if you get a court order but then you'd have to argue in court a reason as to why they should remove it and unless there's an imminent danger to you I can't see them granting it. 

The vigilante websites and others of that ilk I wouldn't go near with a barge pole. Nothing good will ever come from interacting with them. 

The thing about multiple search terms seems to have been based on a misunderstanding of what to submit to Google. If you Google yourself and submit the results page to Google, they will remove the URL for that page from search results, not the individual pages. If you put each individual URL in the application and the application is approved, all those URLs will be removed from search results. After that, if you set up a Google Alert on your name, Google will do the searching for you and let you know if anything comes up. You can then reply to your original notification from Google with the new URL(s) and they will remove those links from search results as well.

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If you are to punish a man retributively you must injure him. If you are to reform him you must improve him. And men are not improved by injuries. (George Bernard Shaw)