Ours isn’t a public lesson barn - the only riders at our place are owners, so yes we are open with the same restrictions we’ve had in place since the lockdown last spring ended: booking time slots to ensure no more than 5 people on property, no family members or guests, social distancing at all times, doors and windows all open for ventilation, riding outside as much as possible, etc.
The wording even in this new stay home order and restrictions is very clear that owners (and their representatives which seems to include leasers and part boarders) are allowed to visit, care for, and ride their horses. And people whose job is training horses are allowed to continue working.
That being said, it is entirely up to the individual barn owner what they are comfortable with. I know a couple that, for their own health and safety are not allowing boarders in the property at all, and an another one that is only allowing boarders one day a week. I don’t think there’s any point in trying to convince a barn owner to change their rules if this is the direction they have decided to take. Your choices may be to simply accept it and wait it out, or move your horse.
The grey area is group lessons on school horses. For that it seems to depend entirely who you speak to at your regional health authority. Some barns are being told it’s allowed, because it falls under animal training. Others are being told it’s not allowed because it’s a group recreational class. In the first lockdown last year there were so many people reporting barns not following the rules. Now I think most people have adopted a “don’t ask don’t tell” policy.