Unfortunately, a flyback requires ratings for a transistor waaaaay above what you would niavely expect.
The venerable old 2N3055 is specified to only 60v on the collector. The TIP35C mentioned in the video is 100V rating, but even that is rather low for the suggested configuration.
When you operate a flyback, the rise in collector voltage is determined by the energy stored in the primary inductance, and the capacitive load connected to it. It's not uncommon for TV flybacks, and auto ignition coils, to develop 300v on the primary, and therefore the collector of the driving transistor, when driven from a 12v supply.
It helps to measure the collector voltage with an oscilloscope, as the power supply voltage is raised.
There are several things you can do to avoid failure due to high collector voltage ...
a) Use a transistor with a higher voltage rating
b) Increase the load capacitance, to absorb the inductive energy at a lower voltage
c) Use an overvoltage clamp diode collector to emitter (inefficient)
... but it's all a bit hit and miss without an oscilloscope, so you can actually see what's happening.
You may get shown videos of 2n3055sTIP35Cs working in this configuration, but if their transformer is a bit soggier, or their transistor is a bit more robust, theirs could work OK, and yours fail.
The conductivity being only two thirds of that of copper wire of the same cross sectional area shouldn't be a problem, especially given a very low current application like a tesla coil secondary.
Enamelled aluminum wire will thus be just fine electrically, but mechanically it is problematic: It won't wet with normal solder, be it lead free or not, unless you actually have copper clad aluminum wire. There are some fancy solder/flux combinations that strip the oxide layer and bond with aluminum, but they aren't exactly common. Even crimp connections of AL with copper/brass fittings are problematic, as such joints tend to corrode, but they should still be your best bet in this case. Another option is to terminate the AL wire with conductive epoxy.
Best Answer
Have you tried finding the current through the coil? Find the current through the coil. You can either simulate it with a spice package or place a meter between the coil and transistor. Once you know the current use the equations located in this online textbook in chapter 11 Inductance and Magnetic energy (Yeah it requires math).
Then find the magnetic field through the coil (you'll need to know the area of the coil and the turns.) Then calculate what kind of energy you need at the input coil. I'll give you a hint as to what you'll find. Your coil is generating a magnetic field at a certain frequency, keep in mind the magnetic field is going to 'attenuate' by the the inverse cube of the distance (magnetic field=1/(distance^3)). Meaning if you had 1 unit of magnetic field at 1cm, at 2cm you will have 1/8th of the field, at 3cm you will have 1/27th and so on. By the time you get to 10cm you will have 1/1000 of your original field so you will need ~1000x the current or you will need to redesign your coil.
This is also why there will forever be companies that promise wireless power but you never see any products beyond the charging pads. The lightbulb demonstrations use phenomenal amounts of power even with a directed antenna(coil). The best one I've seen is this one Wireless power transmission. You can changed the antenna/coil however and it can help. Look up Helmholtz coil if you think that might work for your application. If you want you can parallel several BJT's if you want them to get less hot.