I did a similar thing with my redneck Sous Vide cooker. I used one of those cheap DS18B20 temp sensors in my crock pot, and one of those cheap relay modules to turn it on and off.
I thought I might need a fancy schmancy PID algorithm to keep it close to the set temp, but I was pleasantly surprised. With just a simple "turn on if it gets below the set point and turn off it it goes above the set point" algorithm, it keeps the temp within .5 degrees F above the set point and 1 degree below. Cycle times are anywhere between 1 and 4 minutes.
Now granted my crock pot is only 150 watts, and it's full of water so it takes a while to see the results after you toggle the power. But even my cheap relay is rated for 10 amps at 125 or 250 volts AC, and that's within your 1kW range. There are a couple of issues with using relays for pulses. How long are these "pulses" going to be? My pulses are 1-5 minutes long, and that works fine. But mechanical relays are mechanical devices and have a limited number of times they can cycle. Do you care if your relay is clicking away at 2 or three times a second?
Mechanical relays are also relatively slow. Mine says 10 ms. There's no way you can catch a zero crossing or anything like it at those speeds. So you would be limited in what you can do.