Assuming the foreign judgement passes various criteria (e.g. not being repugnant to the constution) it's essentially treated as a contract to make the payment-- as if you had some agreement to pay the other party, and enforced as such.
Feedback I've heard ranges from "almost always" to "typically", but I've never seen actual stats on it. It's going to be pretty fact specific, so unless your case maps well to a common fact pattern any statistic may not be too useful.
It's common enough in any case that several state's passed laws to deal with foreign (particularly UK) defamation lawsuits, and then a federal law. So a specific kind of abuse of this process was common enough for a federal law on the subject.
Sadly it seems we lacked the foresight to realize that to the extent that many of these were false and malicious cases that the parties can just cook up some other grounds. So when Wright sued critics in Norway, the UK, and Antigua he did so on the basis of defamation, but when his target list included US persons he didn't use a defamation claim.
> If you're in the UK and I'm in the US and I sue you, it doesn't seem reasonable to expect you to have to travel here or to spend money on local representation to defend yourself.
So one of the reasons to deny domestication is that the foreign court didn't properly have jurisdiction over the subject matter. If I had no connection to the US then there wouldn't be a reasonable basis to claim the US court had jurisdiction over me.
But if you and I had some interaction then on the basis of that interaction I might have subjected myself to your jurisdiction. The internet has made it easy to arguably be subject to any jurisdiction anywhere in the world. If you look at some of the arguments in English courts (for example) on jurisdiction they like to emphasize e.g. that someone publishes in English documents which are available and read in the UK. There is a bar for jurisdiction, it's a real bar, but it's not necessarily a particularly high bar.
To complicate matters, summary judgements and other pre-trial judgements like jurisdiction are generally decided on a basis of assuming the claimant's facts are all true. This is a real problem with litigating with an unashamed liar who is primarily interested in damaging you with the process, since they can just make up whatever facts they need to keep the case going.
You don't have to convince me that the existing process is not good. I think it made more sense decades ago when it was much harder for an ordinary person to be arguably subject to arbitrary jurisdictions.
Aside, NY does have domestication rules too, though it does sound like they're somewhat better than the more common version. (e.g. safer to take a default judgement)-- though I haven't really looked into them.