I'm going to guess that their system was something different at core (like electric) or something very wrong with the set-up.

For yours I guess others have laid it out - the temperature of the water flowing through should be moderate. I've had a mixed system of rads and floor, and quite clearly distinctly different loops and temperature settings (note some rads can be designed for low temperature).

For our radiant flooring, the heated flooring sections are never cold - they move slowly in heat terms, so even if the heat was cut off it would take hours to get to 'cold'.

The underfloor heating - was this electric set in under the tile, or liquid from a boiler?

If it's from a boiler, sounds like something set up incorrectly to me.

If it's electric as others have suggested - should be set to some lower output level and to come off and on more frequently, so that the tiles at least stay warm. Is this underfloor the only source of heat in the room?