Yes logic says its the radiator. Fact it gets to and hold temp for a while says to me the thermostat opens and introduces more cold water,until it gets hot. Also as I can drive it by controlling the heater and heater fan tells me the pump works.I had this problem just a few months ago I’d changed pump thermostat viscous fan and hub flushed system through and back flushed. I took radiator out flushed separate and back flushed. It would drive like you said then the temp suddenly climbed. It turned out to be a blocked radiator core. If you get a new rad go for the upgrade one from euro car parts the core has bigger tubes on than original one. It stopped my problem straight away but make sure you need is properly.
So I have the new radiator, but as I don't want to go back on it again I am changing the thermostat and pump at the same time. All three have come to under £100 total so not bad. Just waiting on the pump to arrive.