Infact some used to .. eg. Manchester.
However, in the UK it was decided that allowing this restricted competition and fair trading.
Airlines complained to UK govt. that being forced to use the airport only agent restricted the market/kept prices high and was an unfair restriction of trade.
Airports were forced to allow licencing of ground handling agents, they could compete with their own ground handling if they wanted, but this had to be at "arms length" from the main company.
I think some of the smaller airports might have been given an exemption from this.