If that's wot you are doing, wot Andy said!
And it is actually worth undercutting the scribe slightly - gives you an apparently better fit, as it can crush a little. You often see this on old skrtings ("baseboards" to the Americans), and it's how I was taught to do them 30 years ago.
The >110-year-old picture rails I've dismantled recently actually had inside mitres, caulking and a lot of paint over them. This was a bit surprising, as several alcove inside corners definitely weren't square.
E.
PS: I was taught to pull cut, too. My (also Stanley) saw has only the bow for blade tension, i.e. no actual adjustment of that - you get what you get (and you do up the clamp completely at the handle end). I have tried push cutting with it once or twice and the blade sprang out at the handle end (it's in compression).
You might also experiment with a Japanese saw for mitres - the tooth profiles are wonderful, and it will prove to you that pull saw technique can yield wonderful results too. Obviously an ordinary Japanese saw won't cut curves, but it's a similar technique to a fretsaw, using different muscles from those used for a Western saw, and having a very different feel. I must confess that now I have a couple of good Japanese blades (only rip and crosscut - nothing too exotic), I hardly ever use my Western saws by choice - this has surprised me (in a good way!).
That said, I have really struggled to find good quality fretsaw blades affordably here, and that is an issue. There's a trade-off: a lot of set with respect to the blade's front-back thickness makes it easy to manoeuvre, but also makes it hard to track straight. a deeper blade (WRT the set - the converse of the first sort) will cut straighter, but can only turn in a larger radius. It's just the same as a bandsaw or scroll saw blade. Some have even been blunt from the packet, too. I'd check whatever you use is sharp and properly set, and not asssume.
I won't be the first person you've heard or read complaining about modern Stanley quality...