I will do this the once, then I'm out. In your example.
The driving instructor has a turnover of £3,125 per month and expenses of £800 per month (assuming that covers petrol, wear and tear and all other sundries)
This leaves a Profit/Income of £2,325
The Government is going to pay 80% of that - £1,860
The PAYE employee is getting income of £3,125 per month and expenses of £0.
This leaves Income/Profit of £3,125
The Government is going to pay 80% of that - £2,500
The PAYE employee will get more from the Government, because his income is greater than the Driving Instructors profit. Whether he spends that income on rent, mortgage or the occasional hour of 'Lawrence Dallaglio' is neither here nor there
If the PAYE's Income was the same as the Driving Instructors Profit, the Government would pay them the same, but in your example the PAYE person had a larger Income/Profit to start with.
And, if I where you, I would stick with the AA and not go independent