I'd keep the brakes because they're good... I use a single Weinmann centre pull in my vintage fixed gear and it provides more than enough stopping power.
Alloy wheels will really help for wet braking and lighten the bike as will replacing the cottered axle with a #5 tapered spindle and more modern crank... if the cups are good just keep them.
The wheels may prove to be the priciest item.
Any freewheel with more than 28 teeth will require a long cage derailer.