Yes, this is possible. Not a stupid idea at all. A lot of students and people in cities with very good 4G coverage do this.
However... you should be aware of the following:
- O2 might choose to block tethering
- Make sure you really can get 4G and decent data transfer speeds before cancelling your BT broadband
- 25GB/month isn't very much for household use
- Make sure you phone can provide WiFi coverage for your entire house
- Choose a fixed location for the phone. You might not want to keep it in your pocket, it will probably get hot!
- If your phone is a Samsung Galaxy Note 7 you might want to rethink your entire plan, install a fire suppression system, or at the very least, increase your fire insurance coverage.
If you use Netflix or other video streaming, 25GB
really isn't very much. If you know your own data usage pattern, then you have something to work from. If you don't... I use about 57GB/month (averaged over last 2 years), but it has been as high as 120GB over Christmas. Will O2 permit you to exceed your data allowance by that much - even if they charge you?