We travel a lot with our Healey but we only carry a few spares that don't take up much space. For example: all new spare ignition parts, fuel pump, extra wire, digital multimeter, nylon tow rope, shop manual, flashlight(s), Moss or Victoria British catalog, Austin Healey Club membership roster(s), old copy of Hemmings Motor News. I do have a spare head gasket we have been carrying around for years, never have needed it but it doesn't take much space
We also travel with a laptop that has been very helpful for tracking down parts and suppliers. Yes, we have a cell phone and AAA RV plus (that's 100 miles of towing).
We try to leave with the car in good condition, we fix stuff that we think is marginal before we leave. .
We have had problems on trips, but mostly minor ones that don't need attention right away or that we can fix on the road ourselves. Carrying a mountain of spare parts is not a good idea. It will jam your Healey with extra weight and what fails will not be in your spares cache, Murphy's law applies ere.