For a turbine blade, you'd use a single-crystal superalloy. The machining is just 3D printing of wax; you then use a lost-wax process to cast it. Cooling channels are built into it.
That ought to work. You don't even need a particularly corrosion-resistant superalloy if you cool it with liquid methane and then let that leak out to form a moving film on the surface.
One thing to keep in mind about spike nozzle cooling is that you get compression shocks from the aerospike flowfield that strike the centerbody. These shock impingements drive up local heat transfer rates. As the flowfield changes with increasing altitude (decreasing ambient pressure), the impingement locations move. So when you size your cooling capacity, you have to account for this which tends to make the spike "overcooled" as compared to what you'd have to design for with the bell. This tends to correlate with increased pressure losses from the associated high coolant flow rates, causing a system-level mass hit and detracting from the nozzle Isp efficiency benefit. Take a look at XRS-2200 test video to watch the spike ramps ice over hard after engine shutdown. While not unique to that engine, the magnitude of that effect has a lot to do with how heavily cooled the ramps were.
How would you build cooling channels into something like this?
Picture of blade and some process description: https://en.wikipedia.org/wiki/Turbine_blade
Picture of wax blade without cooling channels: https://en.wikipedia.org/wiki/Investment_casting
Company that uses DLP 3D printing for turbine blades: https://www.prodways.com/en/lost-wax-casting-of-turbine-blad...
[1]: https://www.relativityspace.com/
[EDIT] If you're reading this and you work at Relativity, I'd love to know if you actually 3D print your nozzles...
The nozzle must be made out of graphite, the manufacturing of which is much more complicated than aluminum, so throw out the door any pre-conceived notions you may have that making any shape is possible. Our nozzle was a three-part component: the inner plug (think of one of those "nose suckers" you have to get boogers out of a baby's nose) and the outer nest in which the plug sat. Between the nest and the plug you have "struts" supporting the plug to allow for a gap (through which your fuel will be exhausted). The plug is secured to the nest via some high-heat epoxy.
So yeah, way more complicated than a single bell of graphite, which is probably the #1 reason why these haven't seen industry use (the cost of complexity must be weighed against the cost of inefficiency, both are important factors in any rocket application).
I am very interested in the "linear aerospike"[1] engine though, which would obviously have design implications on the rest of the spacecraft, but manufacturing a linear aerospike would be much simpler than what we attempted.
[1] https://en.wikipedia.org/wiki/Aerospike_engine [The main image]