I dunno. When I play against PA, I'm usually like "Okay, what's PA up to?" every few seconds and of course check the creep lines. The minimap aspect of Blur is great, don't get me wrong, but I think it's really just going to delay reactions of people looking for where she is farming on the map, and of course in the middle of a fight, your attention is divided. If she isn't there immediately, you have that unknown factor is she is coming out of the trees or on the other side of the map farming. I like the idea of it, but I don't think it makes her a good split pusher. Pubs do usually care about towers going down. 1600 range is decent, but keep in mind it isn't that far off of Blink range. In many cases, if you see that icon go away, you might be already dead.
I think her Phantom Strike not being a real escape also makes her weaker in that regard. At that point, why pick PA to do that when you could pick AM and do it better?
At the risk of having this conversation always turn out the way it does...
Look Hylian. I'm not saying PA is a better split pusher than AM. I'm not saying PA is the world's greatest carry or even Spectre tier.
I'm saying: You think PA is crap with a Battle Fury, partly because she can't even split push like AM can. I'm saying you underestimate her ability to split push with Battle Fury, because of Blur. Note that this is not saying that she is a top 10 split pusher, but that she's not junk at it.
What might a hero who is good at split pushing have? A rough list might include:
1. Ability to clear waves quickly to get to towers (Battle Fury, check)
2. Physical damage to take down buildings quickly ( Battle Fury adds a nice bit of damage. Some even build Desolator on PA as a fighting build. Even without Deso since PA builds for damage as a hard carry she's about even with this with AM etc )
3. Ability to escape when defending team comes back to save towers ( Blink Strike's not anywhere good as just Blink, but it's better than heroes without any kind of innate ability. Blur also helps with this since it's an early warning system for you. )
Blur's minimap invisibility adds a 4th element that's pretty much unique to her: the surprise factor. No other hero has the this "Oh what how did that tower go down so quickly it should just be creeps?" factor. Again, maybe it's not going to work at pro-level games but at 1300 MMR (which, bear in mind, is the player you're giving advice to) it's probably going to work well every so often.
So. Circling back to the original point: Why is Battle Fury not absolute junk on PA? Because it gives gives her farm, damage, sustain, and some ability to split push. Note how these are all good things. Maybe EE and Balto Avanta both prefer the "MoM Fight early" build for PA, but just like how some people prefer a Radiance-Octarine build on Alchemist and some prefer a Treads-Medallion fighting build, the Battle Fury build on PA isn't irrelevant.