This ages me, did South Africa 1997 with Gullivers so not sure how relevant to now as was probably the last amateur ethos and non commercialised Lions tour, so much so that we were in the same hotel as the Lions in Joburg for the 3rd test, had some amazing nights with them all paid for on Fran Cotton's room number which they were giving out. I did go back again DIY to SA to watch cricket in Cape Town over the Millennium new year as well.
In a different place in life now to then and any trip now, I am sure like yourself, the priority would be making sure whatever we did the kids were happy as that generally leads to happy Mum and Dad!
There will be people on here who have much more recent experience of SA than me and it is difficult without knowing any schedule / venues but my feeling based on no additional research whatsoever is that if I was taking my kids to SA I would probably go organised tour. Depending on the kids age there may be bits of it which they might not be so keen on or are not necessarily most suitable but the guaranteed tickets, decent accommodation / transport, additional events such as dinners / pre match venues and a Company rep to go to if you do have any problems or need advice would probably be my choice, however that of course does cost!
Feeling jealous as well as old now!