Boiler only runs for 4 minutes then shuts off

I feel your pain, been there myself. This could be a problem with your boiler, it could be your system is gunked up, or it could be your thermostat commanding the heating system to turn on and off. My money would be the boiler and not the thermostat, but you could take the communication wires out of the thermostat and twist them together for awhile to see if your boiler stays on. If it does then buy a new thermostat and count yourself enormously lucky that it was that simple and cheap.

It could be that your heating system needs a flush, which is something you could do yourself with basic tools and some hose. Generally you would flush the system with water, then refill and add a cleaning compound for a couple of weeks, and then flush again and refill being sure to add a conditioner. If you live in the UK then these instructions may work for you. If you're not sure what you're doing then get a professional.

If it is your boiler and not your thermostat then your best option is to get in a very good heating engineer who really knows your boiler type well. There are simply too many boiler types and things that could be going wrong to give you advice, and you probably wouldn't be able to fix it on your own anyway.

Your goals are somewhat at odds. Even where radiant floor heating is used as the only source of heat, the floor typically does not get anywhere near "toasty." The floor temp is typically only 5 to 10 degrees above ambient (75 to 80 F), and the vast size of the radiator (i.e. the entire floor) provides enough energy to heat the entire space with such a small differential. So, the floor will feel better than an unheated floor, but since it is still colder than your body temp, it won't feel "toasty" by any means. So, if you piped enough energy into your floors to make them feel warm under foot, you'd likely overheat the house!

The only "toasty" floors I've experienced is a heated bathroom floor. In that case, there was electric coils under the tile (several hundred watts worth for a large bathroom), and indeed, the floor felt warm underfoot.