They're hard on cars. Birds seem to search out freshly washed ones to poop out their mulberry feasts on.

"Especially wondering how it will affect the soil ph, considering how sensitive hydrangeas are"

Fallen mulberries are not going to have any effect on soil pH. Even if they did, hydrangeas are really not sensitive to soil pH at all. Soil pH does change the colour of the blue/pink ones though...blue=acidic & pink=alkaline. Whatever colour they are, the mulberry won't change things.

LOL calliope.
I have the misfortune of having one but as low-grade trees go, they could be worse IMHO. They don't spread like black locusts, and their roots are much easier to cut (although it's all relative) than junk maples. What I would worry about in OK is that my guess is they are pretty thristy compared to some other deciduous trees, but I have no idea whether that's ever been scientifically quantified across many commonly planted trees.

The birds will also replant then around your yard ...

This year I froze enough mulberries for several pies. The tree offers shade from the Texas sun and along with several other trees, helps to keep my electric bill down as I use less AC due to the shade.
It does not overhang the driveway, and the shrubs, grass and roses do not appear to be affected by dropped mulberries, perhaps because squirrels and birds consume a lot of them.

On the upside, my grandma made Rhubarb Mulberry Jam that was absolutely to die for. Even richer tasting than Blackberry jam, my second favorite.

Aaaargh!!!! The mulberries are much more plentiful this year than in years past, no doubt due to my watering of the new garden underneath. The birds and my dog haven't eaten up enough of the berries, and with the heavy rainfall we've gotten lately, the mulberries are rotting and molding on the ground! I can't even stand the stench long enough to weed the area!