Hi Rufus
I've done this type of repair hundreds of times and would go with the scarf joint Inspector recommends but I always add two side by side biscuits (size 0 or 10) along the scarf, being careful not to let the biscuit poke through the leg. This helps the joint carry the sitters weight and stops the joint sliding when clamping. Try to match the growth rings of the new part to the legs, it looks better but only you will notice.
There could be a problem doing this as the stretchers are quite low to the floor and the rot may go further up than shows in the photo.
If the chairs have been somewhere damp the joints might be coming lose (the right seat rail in the photo looks to be opening up) It might be worth knocking the rear seat and stretcher rails apart to get the back section off to make it easier to work on the legs.
If it's not possible to do a scarf joint due to the stretchers you might have to add a block to the tip. This is not easy but can work and is the way I would do it on turned legs. Cut back to sound wood and get an absolute 90 degree clean surface. Make a drill guide by making a hardwood piece the same size as the leg and about 40 to 50 mm long. On a pillar drill make a 12 or 15 mm centred hole through the guide piece. Glue 4 or 6 mm ply on the guide sides with about 75 mm extensions to position down the leg, clamp this on the leg, making sure it is tight to the leg cut surface, and drill to around 35 mm depth on the leg. Make a 2 or 3 mm oversize blank for the tip and turn on a lathe a pin the right size for the hole, leaving the rest square. Aim for a smooth fit, not slack but not too tight. If there is enough room I sometimes add 6 mm dowels towards the corners of the new part. On the pillar drill make four 6 mm holes near the corners but not too close. Again check the growth rings on the leg and new piece, mark the position you pick on the parts. In the 6 mm holes put dowel points in the new piece and push and tap on the end of the leg. You should then see four indentations on the leg where you can drill 6 mm holes with a lip and spur drill. This isn't easy to get spot on so you may have to pare down the dowels to 5.5 mm. The dowels only need to be about 30 mm long. Glue and clamp then pare down the surface. If the back legs curve you will need to make a block to clamp along the leg to get an accurate clamp orientation.
Whatever way you do it it has to be done very carefully and accurate. Get it right first time as there's no second go.