Did someone call?
The happy truth is that you cannot meet all your drilling needs with just one tool, but what follows is all about using tools, not collecting them, honest!
The key difference is in the chuck. For the 'old style' bits (which for some odd reason, you seem not to like!) the square shape tang was often gripped by a two-jaw 'alligator' style chuck. That is what is normally fitted to a carpenter's brace. Some later models have a groove along the jaws which will also grip round-shanked bits bigger than about 5mm, up to about 13mm. I have a nice solid 'Toga' brand like this. Pretty much any Stanley with plastic handles will also have this sort of chuck. Look for anything from the last 50 years or so.
Here's a picture of the works of an old alligator-jawed chuck, with a groove!
And here are the bits of a three jaw chuck which will only grip round (or hexagonal) shafts:
As well as a brace, you'll need a breast/belly drill for big holes. There are plenty about in good nick with two speed bevel gears, easily swapped. I'd go for Stanley, or Chapman, or indeed Millers Falls. Try and find one which is complete with its side handle; that's essential. Some of these were offered with a choice of two jaw chuck (for the old square bits) or a three jaw chuck - you want the three jaw. It will normally be fatter, and may need a key.
And then you will need something small for pilot holes etc. These were a popular diy tool so there have been cheap and nasty copies around for years. Price is not now any guide to quality (at least, not on eBay, while these remain under-appreciated). Look for a chuck which closes up nice and smooth and decent wood on the handles.
Better ones have a second small bevel gear which doesn't drive anything but reduces slop in the mechanism. The nice old Millers Falls ones have a cute flanged jockey wheel that you can use to reduce slop entirely, provided your gear casting is true.
So, Happy hunting! And thanks for giving me a chance to post this gratuitous snap of 'braces corner' for the first time: