I have a reasonably varied experience of HE - I have HEd my DS (through which i met a number of other HE families); I have taught children who have left school to be 'home educated'; and i have taught children coming into school having been HEd for the early years of their education.
IME, the success of HE depends very much on the reason for HE and the specific parent / child combination. Simply because it is so individual, it is MUCH more variable, both in terms of process and outcome, than standard state schooling. There are great successes, and theer are great failures, and in between a range of those who end up following round about the same path as they would have done had they been schooled, just via a different route.
I HEd because my DS was damaged by being in a particular school, to the point where he was a selective mute being investigated for ASD. My initial target was to HE him to get him ready for a school transfer we were anyway doing due to a house move, but if that wasn't successful, no further than the end of primary. My reasons for that? DS was - still is - spiky in profile, and while some of his strengths coincided with mine, others would, by the end of primary, undoubtably have reached a point where he would benefit from being taught by a specialist in that field. As it happens, HE mended him and he returned to a very different school, where he thrived.
Those children I taught who left school to be HE? They were from the Gypsy/Roma/ Traveller community, who at least where i taught were willing for their girls to attend primary school but did not support them attending mixed secondary schools. Unfortunately, the very low educational attainment of the parents, particularly the mothers - there was a single literate woman on the local settled site, but the rest were sub-literate or wholly illiterate - did mean that the main education received was in preparation for early marriage. While this could be seen to be a success by that community - keeping their particular culture alive and enclosed - I'm not certain that it is a success by any other definition?
The children i have taught who have entered mainstream primary schooling from HE have, without exception, been very far behind their schooled peers. This does not mean that HE doesn't work - just that it hadn't worked for those families, which was probably why the children were then entered into school. Interestingly, the problem was absolutely not socialisation, or SEN - just that the methods used to try to teach them had not succeeded up to that point (a child who could not read despite parental efforts but had never been taught phonics, for example, learned exceptionally rapidly with systematic phonics teaching, a method unknown to their parents). I am not suggesting that such children are in any way representative of the HE community - those who make good progress in HE will tend not to enter school.