Your house is too large for adequate coverage by a single device unless it has no interior walls and you don't care about the speeds as you get further from the wireless router.
You should be looking for a router without WiFi but with proper QoS. Your WiFi can then be done with multiple AP's to provide better coverage throughout the house. You just need cables from your switch to the locations where you'd want to put the AP's. The solution to bad WiFi is not a more powerful transmitter. You need to reduce the distance to each WiFi source which you do by adding more AP's. In a case where you can absolute not run cables and nothing currently exists for MoCA then you would use mesh but it'll never be as good as a cabled solution.
If you want coverage to the detached workshop, run a cable there and add an AP. If you can't run the cable then use a wireless bridge designed for point to point and then add an AP to that.