As someone who runs an events business that regularly organises to resell bulk quantities of rooms with hotels, I can offer one reason colloquially from experience that builds off other answers:
While other people have identified that reception staff may not have the capacity to actually change the price to match booking.com, they may want you to use the website so that the room is booked under the block of rooms available to booking.com at that price. Yes, you can argue this technically goes under the umbrella of "accounting/bookkeeping reasons" but it's very likely that there is an agreement between the hotel and booking.com that X number of Y type of rooms will be able to be sold at Z rate (discounted) for a certain date range - they probably don't want to sell all the rooms in the hotel at that (probably) low rate. The reason the booking.com deal exists in the first place is for various advertising reasons brought up in other answers.
In practice, it's likely that there will be a cap on the number of rooms for a certain date range booking.com can sell at that price. If the front desk matched the rate, that wouldn't necessarily decrease the number of rooms at reduced price available to booking.com, resulting in an overall reduction in revenue for the hotel in a scenario where the marginal room-night was going to be sold anyway.
So for example in a figurative tiny hotel with 15 rooms available on 1 night (scale up to real life in your head as necessary), where booking.com had the rights to sell 5 of those rooms at USD50 per night, whereas the normal price was USD100 per night - in the scenario where the front desk matches the price, and all rooms are sold, the hotel loses USD50 in revenue (because after selling you a USD50 room, they still have to let booking.com sell 5 USD50 rooms). Whereas if they tell you to use booking.com, your USD50 room is one of the 5 sold through booking.com and if enough rooms are sold, the hotel does not miss out on revenue.
The potential follow up of "but why can't the front desk just count it against booking.com's block" is some combination of communicating with people is hard/time expensive, the contract may or may not allow it, and again non (upper?) management staff may not have the capacity.