if I want to reduce this to file handle size, would I do better to try to square it off with my 10" table saw or try to split vaguely square pieces and round them off for final use?
I'm sure it doesn't matter. For something as low-stress — meant in both ways — as this the wood you use, and how it's derived from the original stock, is not that critical.
Heck, even the species doesn't matter much! Perfectly usable handles can be made from wood waaay down the food chain from black cherry.
My friend, who's opinion I normally respect, informs me that I have to split match the grain.
The conventional wisdom about splitting (AKA riving, hence riving knife) yielding stock of superior strength has a pinch of truth to it but TBH once you start to do any surface dressing — and creating a handle, of any shape, would certainly quality — any original advantage can completely go away. Plus, as already mentioned, file handles are a low-stress application anyway.
About a decade ago in another woodworking venue I brought up this point, that the advantage of riven stock was largely a myth, and got resoundingly dumped on by most responders (as I expected to be) but it doesn't take a rocket surgeon to think through what happens when you move away from the riven surfaces as you further process the wood. And this happens swiftly with almost any shaping, even the merest intervention such as forming a board where in only step 1 or 2 you make one face parallel with the next!
Now if you're doing a tapered or complex-shaped handle the same goes in spades since it's possible to likely that you retain none of the original riven surfaces. Ergo, zero benefit was gained from splitting the wood out in the first place, unless you count safety1.
My experience is that when I split the wood, it breaks up the grain away from the split. When splitting logs, I would chop into the end of the log two or three times to "break up the grain" and after that, I could start seriously splitting.
You're splitting in a different way to how I imagine your friend is envisaging, where you quite carefully choose the riving planes. When you're just splitting the wood to break it into fireplace-friendly chunks you may completely ignore those niceties.
Arguably this is the defining difference between splitting and riving, although the words do seem to be completely interchangeable in common usage.
I am trying to see about using some of the bigger pieces to make small things, such as file handles.
You don't have to just be eyeing the bigger pieces. Not at all.
Don't overlook the....
Other usable wood from any tree
Some parts of a tree that would normally be considered junk wood from a woodworking perspective — not just only fit for the burn pile but destined for the compost pile or wood chipper! — are not just viable for small tool handles but perhaps even preferential. One reason would be that it allows you to retain more of the bigger pieces for non-utility projects, such as jewellery boxes, pencil cases, a Shaker candle sconce, maybe a rack for these files you're handling.
I've made a boatload of tool handles over the years and they can sidestep a lot of the (otherwise perfectly sound) rules of thumb about good wood simply because of the small scale, both the cross-sectional area and very much so the relatively short or very short overall length.
These other parts include the branches, including their centres with the pith if you want2. But honestly, nearly anything is up for grabs here, including any pieces with sapwood, root stock, and branches that are a mere 1 1/4" or 29-30mm in diameter, all can be perfectly reasonable. Short stock too, blanks as short as ~5" / 127mm are fine as long as there's minimal or no checking.
A note on drying/seasoning
I hope you realise you're not going to be using the wood any time soon? Like not this year.
That is unless you force-dry it by heating in some way (solar kiln, toaster oven, microwave) or you do something even more unconventional3.
Also, it's vital to seal the end grain pronto if you want to minimise checking and maximise yield, especially from shorter pieces. Hands down the best way to do this at home is with melted wax. As you can use up any scraps of wax (regardless if sooty or dirty) the method is cheap to nearly free, extremely effective and fast.
1 Which is not an insignificant aspect of this, given the difficulties in safely sawing up chunks of tree that are irregularly shaped and prone to wanting to twist or rotate when being power sawn (totally different picture if sawn up by hand).
2 If you centre the pith (ideally perfectly) it's great because then the wood is pretty much equally strong in all directions. But as I've found out empirically during my travels having the pith slightly off-centre, esp. at the butt end of a handle, honestly doesn't matter too much except aesthetically..... I do love a perfectly centred pith at the visible end of a handle :-)
3 See Something else to try, in this previous Answer.
