If you live near a local engineering firm, someone making wrought iron gates or even a local blacksmith or farrier, then the bit that you want can be easily made.
If you don't, then all is not lost, as I've seen this type of gate resting in a concrete 'cup' that was made by the brickie rebuilding the gate pier - this consisted of a piece of tube cut to the required length with an internal diameter the size of the spigot on the bottom of the gate stile - this was set into the wet concrete and well greased before the gate was re-fitted.
Wouldn't worry, haven't often seen one of those that wasn't badly worn egg-shaped.
Mine are car bits scrap (CV joint? roller bearing race?) You need anything vaguely conically cupped for self-centring behaviour and about the right size. Hole in the bottom for drainage. A taper also lets you adjust the effective diameter by shimming it up & down.
Otherwise a lump of random cast iron (wrought iron if you have it) and a lathe. It's more rust resistant than most steels. If you're not worried about impact damage, winebottle bases make a great outdoor bearing (traditional for weathervanes)
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