I can buy a whole-house access control system based on business use for that price. Maglocks on the doors, control units, RFID readers, PIN-pad entry, the works.
Even on a cobble-together budget with those products you're looking at over a grand before you even start fitting.
Get a couple of mag-locks for your gates, and the various entry locks (they normally just replace the lock on the door-frame side of the lock) for the house, and a simple controller (there's a PIN-pad and RFID - in whatever combination you like - one on Amazon for £10 with 12V relay control...). You've already had to build a Fritzbox and run a relay cable to remote control it, so you haven't lost any functionality.
And, my biggest question about doing all this (as I have the parts and the know-how and have considered doing it on my house) is what about the insurance? House insurance typically demands a certain standard of lock and access control does confuse things on non-business premises.
Hell, I had SMS / telephone-controlled relays rebooting the ADSL modems at my previous employer - one 3G stick, a pre-pay SIM, one physical "button", one Velleman K8055, an old car relay and anything you can be bothered to put on the other side of it. My employer used to be able to reboot the modems with a text message (with customisable PIN, or other commands!) from anywhere in the when the VPN fell over.
Either cobble-together cheap components, or buy an expensive all-in-one solution. Don't cobble together expensive components.