It barely worked that time. It wouldn't work now, because the sort of people who can spend £100,000 on the first year of karting can't do that by dividing their attention between multiple jobs. (I think Anthony Hamilton himself concentrated on one job, once his IT consultancy took off).
I think it is theoretically possible for a non-millionaire to make it to F1, but it would have to be through the parents (or another close relative) having that £100 k available upfront and then being so amazing that a rich sponsor or three covers the rest. (Most likely, that would require impressing the right manager, although a family with the right contacts might not need them). Certainly, nobody middle-class or working-class who doesn't have a great manager or rich sponsor in their contact list need apply.