Unless there is manufacturer support, this has never been the past.
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That's because in every other major motorsports the rules are usually made by some other government body/organization than FIA. In Le Mans/WEC it's ACO and IMSA, in Dakar/W2RC it's ASO, in F1 it's FOM. These don't really care about manufacturers (it's ofc more complicated in F1, but they still allow privateers).
In rally it's all about FIA without any other outside force. All they care about is the manufacturer's money. I don't know how much do current manufacturers pay for homologations and stuff, but I guess that if they allow privateers at any level, they could get same or even more money by them from homologation fees. Don't forget that there are only 6 manufacturers now that currently make rally cars - Ford, Toyota, Hyundai, Skoda, Renault and Stellantis. Even if some of the would leave, average fan wouldn't notice as someone would for sure make a car based on these.
But imo to make it work, there have to be more freedom in technical aspect. Both Dakar and Le Mans are pretty free yet cheap.
Since Rally1 cars are not based on actual road cars I can't see why they wouldn't allow private cars now. But not sure it would be viable. It works in F1 because there is constructors prize money and it is commercially succesful enough to attract large sponsorship income.
Even with some support from Ford and Red Bull, M-Sport have struggled.
https://www.youtube.com/watch?v=LQ_KrIlY8eE
5m30: David Richards saying they should've flooded the market with R2's and gone that route