It’s unsurprising DIYLC doesn’t support what you’re doing, since what you’re doing is unconventional. To me it’s not at all clear what advantages there are to your putting components and wires on both sides. I don’t see that you really gain any space — you take up the same pads either way. In your photos I see three resistors mounted on the underside, evidently because if they were on top it would make it harder to connect their leads to pads on the other side of the capacitors. But not at all impossible. As for the wires on the underside, I guess the top side looks less cluttered that way, but I see no reason they could not go on top.
Of course components and traces on both sides is conventional with PCBs, and software like KiCad definitely supports that. There, having traces on both sides is definitely a benefit since traces on the same side can’t cross without connecting; wires can, so that benefit doesn’t apply in your case. Through hole components on both sides is a less useful thing, except for instance putting headers on the back side of a board and pots on the front side. I don’t think that really applies to your case either.
With either PCBs or proto boards, having wires (traces) and components on both sides does have the disadvantage of making it harder to understand the circuit by looking at it. But it’s hard in any case. I get cranky when people say “my project doesn’t work, is there a mistake, here’s the stripboard layout” — it’s 10 times easier to understand and debug a circuit from a schematic. Stripboard or proto board or PCB layouts are to describe implementation; use schematics to describe function.