Are there any legitimate impediments to a business model that relies on investing students' capital, paying them fixed monthly profits, and allowing them to withdraw the principal at any time?

If the intention behind fixing a specific amount for expected profits is to guarantee a known profit for the investor, then it is not permissible. However, if the intention is to determine a percentage of the profit, with the investor receiving a monthly sum on account, and then a final settlement is made when the profit becomes apparent, this is permissible. The fact that you can estimate the profit does not permit guaranteeing a known profit or guaranteeing the capital. Rather, a percentage of the profit (such as one-third or one-half) is agreed upon in the contract, and the owner of the capital bears the loss, if any, provided there is no negligence on the part of the worker.