Has anyone ever seen a temperature controller or single loop controller that has a built in relay that is capable of handling 120V-8amps output?

My gut reaction is that you will not be able to find a controller with that kind of juice on its output. A more realistic option is a teperature controller with a switched output that powers a separate contactor to supply the voltage/current you are looking for.

The more probable failure is on the output. you wouldn't want to change the entire controller for an output failure. would be preferable to change an ssr or contactor instead of the entire controller.

SSR is more reliable and don't have mechanical live expectancy, they're only fear are sparks. and they can handle many amps according with the choice of component you do. And they can cycle very fast.

If you really want 8A you still have the on/off hardware controller that can do it, like oven controller with capillary probe and mechanical contacts. watch out for precision you want. Personally 3A is the more power i know an electronic PID controller can handle.

I came across max. of 3A rating. for 8A, you should use seperate contactor.

A temperature controller "heat" output may control many things: electrical heaters (3- and 1-phase), admission valves for heating steam, or oil, just as the "cool" output can control water admission, air blower, chiller, etc.
Most controllers feature plug-in modules (relays, triacs, etc) for these outputs, and others: alarms for example. Just as for most PLCs, these contacts ratings are pretty low. They are meant to control a contactor, etc.,as per the actual physical configuration.

We have a 1/8 DIN Horizontal mount temperature controller with a 16A relay output option. Normally, we like to use external relays or SSR's for higher current applications. What are you controlling? Is it a resistive load?

Eurotherm had 2 controllers with a 240v 10A relay output but there was no demand and it was obsoleted. These were in the 2000 Series range, a 2404 and a 2204, you may be able to find an old one? Most controllers, certainly in the Eurotherm range, contain relays rated much higher than required at say 6, 8, 10A or more BUT pcb circuitry really can not handle much more than 3A therefore most are rated at 2A for safety sake. Moral of the story.......use a standard relay output and a higher rated external relay or contractor or even better an SSR and keep high currents out of the controller.