Besides doping, thickness is also very important. Of course you can't just wire two diodes together and get a transistor -- there's a bunch of metal inbetween, and metal destroys the semiconductive effects -- but it's still not good enough to take a lump of silicon with two spots on it. Particularly, the base needs to be very thin: under 10µm or so. This is the distance over which free charge carriers (electrons with enough energy to ride in the conduction band, and the holes left behind in the valence band) can diffuse through bulk silicon, occasionally combining along on their way at doping sites (which give off a small excess concentration of electrons or holes, and by reciprocity, also consume them).
So, if you can take two N-type blocks and one very thin P-type sliver, clean their surfaces perfectly, and also polish them atomically flat (I know, so picky, right?) -- you can physically assemble a transistor.
Needless to say, the planar method is far easier, either diffusing dopants in from the top (with depth controlled by heating schedule and concentration), or by growing doped material on top directly (epitaxy, usually using heat or plasma to decompose silicon- and dopant-bearing gasses).
So that's how you make a transistor.

As for the substitute, PN3563 comes to mind, have seen it in plenty of circuits. Though it's lower voltage (and also quite obsolete), which I'm sure Tek also needed for the application. Agreed, the -3571 could be the 2N (JEDEC*) designation, then they modified or selected it from there.
*An industry standards group -- numbers are simply sequential as far as I know, and almost all are deprecated/obsolete. For that matter, I would guess almost all were obsolete within years of their allocation. Probably a lot of numbers were scatter-gunned based on the wide parameter spread of early transistors (particularly in hFE and Vceo or Vcbo), so there was a lot of overlap, too. It's worth noting how this type of standardization works: anyone can produce a "2N3904" that meets or exceeds that spec. A particular manufacturer may give information about their particular part (ON Semi (née Motorola) often does) but that doesn't mean those data apply to all manufacturers. Whereas these days, with many single-source parts, you simply get whatever a e.g. STP6N60M2 is, and if you need a substitute from someone else, you look up whatever crosses to it, or search for something with comparable parameters.
Tim