What I've been trying to do so far, is to separate the completely unsupported from poorly supported. (Not sure if this is the best way, just what I've been doing)
Completely unsupported answers can and should be downvoted, flagged as NAA, and commented to politely ask the user to back-it-up.
Answers that are poorly supported are a little more of a gray area... As you've noticed...
In cases where I've spotted an unsupported answer, and they edit it into a poorly supported answer, I usually leave the downvote, flag, and comment. And optionally add another comment asking for more/better support.
In cases where my first impression of the answer is that it's just poorly supported, I try to comment and optionally downvote. Voting usually depends on my impression of the answer's support... Is this a well intentioned answer written by someone who's stumbling through a learning curve, or is it someone who ought to know better and is just doing the bare minimum?
Now on to what you were really asking about...
In my opinion (pun intended) there isn't a meaningful difference between:
In my experience you should do X...
(Unless, of course the "in my experience" is supported with some explanation of what that experience happens to be later in the answer.)
I don't think this should be sufficient for backing up an answer. It's slightly better than nothing, at least it hints at the idea that the answerer may have done the thing before, but actually explaining and providing an example of what they did and how it worked would be better.
"In my experience" is pretty much another claim that needs some support.
On the other hand...
I'd be cautious about questioning whether a provided personal experience is sufficiently related. And way more cautious about calling people out for making up a personal experience.
I'm sure we'll have people who stretch the limits of credibility in all directions, but unless something seems incredibly incredulous, that's a line I'd rather not cross. Seems to me that may be setting a precedent that may cause all answers to be second guessed in ways that'll very likely turn rude and abusive. We don't want that.
In the hopefully rare cases where it looks like we have a creative writer, it'll probably be better to use a custom mod flag or possibly even a meta post... (Please go with the mod flag first and wait for a response. And please avoid naming and shaming whenever possible.)
Which leads us to another possibility...
What do we do with answers that are well supported by personal experience, but that experience is directly contrary to what the person asking wants to do? These are another gray area...
Some of these may make for really good frame challenges, while others may be just... well... Awful. Still not entirely sure what to do with those awful ones apart from downvoting, but when it sounds like the person answering is answering out of a completely different experience than the person asking... It may, at times, be worth pointing that out.
A frame challenge should still be supported with personal experience and/or references. Perhaps at an even higher standard than our usual answers. Someone writing a good frame challenge should be knowledgeable enough to fully grasp what's being asked about, and be able to demonstrate that knowledge, while being able to support their challenge on top of that.
I don't know anything about X, but you should do Y instead.
Isn't a supported frame challenge. Neither is:
I don't know anything about X, but you should do Y instead. Let me tell you literally everything about Y and why that's a better solution while ignoring X.
These answers sometimes demonstrate a lack of important background knowledge. When X is crucial to the question, it's crucial to address it in the answer. A good frame challenge isn't just a matter of telling someone that they're wrong. it requires some knowledge of why they're wrong, and a certain amount of tact... But tact is probably drifting into what ought to be another meta post.
In my opinion, answers of the form:
I've never been in your situation, and I think your situation is morally repugnant, and you should just stop being that way... But let me tell you all about how people like me view people like you, so that you better understand why you shouldn't be that way.
Aren't likely to be helpful to the person asking. This may seem hyperbolic to some... But we really do get answers that boil down to that. Let's not allow that, regardless of how well supported the answer may be...