Tigger90, Hi.
My Son has a small guitar repair business, from time to time he has to make good a cracked piece of timber, like the neck on a Guitar.
To effect such a repair he bores a very small hole into the crack and using a Syringe he injects glue into the crack.
Suggest you consider the same option? the advantages are that there are no indications of intervention, that is if you bore and inject from the underside of the Handrail a tiny drill hole will not be noticed, the drill hole and the Syringe size are obviously to be of a close tolerance.
The glue used? my son uses tight bond, a glue used by Luthiers, having said that my sons much preferred glue is that old fashioned "Animal Glue2 the one that comes as pellets and has to be boiled up in a water bath. There are several web sites where this type of repair and the tools needed can be obtained, the largest web site is probably? Stew mac in the USA? on that site there are some seriously remarkable repair options and tools are to be found. [other guitar repair web sites are available]
The main advantages of this suggested method of repair is that it is [almost] un-Detectable?
If you do opt to use this or a similar injection procedure? then I would suggest that you bore at least two holes on the underside of the handrail, to each Baluster, make sure the Syringe fits? bore all the holes to all the balusters, then and only then begin injection, this to prevent the glue setting in the Syringe. Finally go back over the Balusters and remove all overspill on several occasions to ensure that the glue you use does not leak and mark the timber?
OK a bit off the wall as a repair option, but ? worth a bit of consideration?
Ken.
Hi my stairs are about 140 years old and I have spent the last month or so removing the paint so I can finish them with Osmo Oil. I have noticed that they are not as stable as they could be, they have a slight flex if pulled and the top of the spindles seems to need a bit of new glue or filler? 
