Lots of problems with that kit. I can't really see an excuse to use a DC converter where there is wall power available. 6418s run fine on 18V or 24V, get a wall wart that outputs that voltage. Or if you demand running on the switcher, why run the filaments off the 30V rail when all they want is 1.25V? Run them off the 5V rail (that also reduces current through R204 a lot, which lets you use a larger value for R204).
But the basic problem is MC34063A, as configured, will have in my estimation >100uV of noise--much more I think. Placed in a circuit with no power supply rejection and lots of gain, that's a recipe for trouble.
At a minimum, the feedback loop (R201-203) should be moved to the right side of R204 (edit: when you do that, reduce R204 to 10, or better yet, replace with a 100uH inductor with that DCR), and a 10nF cap parallel with R203 will help quite a lot. But I see at the end of the article he wisely jettisons the switcher in favor of a regular linear supply.
Also, is it just me, or are C9/C109 ridiculously small? I mean I know LPs don't have a lot of low end, but if you want to filter that you don't make it dependent on load impedance. That's just odd.
Be aware that 6418s will have ~0.1% THD, usually much more. Depending on how the tubes are operated, that could be mostly third or mostly second. If you play with it, you can tailor that as you like. But this is very, very far from a hifi amp, and also far from a vintage tube sound. I dig it, but it won't be everyone's cup of tea.