The problem of a perched water table came to my mind right away. It would be like
growing in pot that is only as deep as where the cardboard is. If you treat it like a big pot (including using a lighter soil mix), you'd probably be okay. But I assumed the cardboard would decompose quickly and it wouldn't matter much anyway. Now I see that the cardboard could last much longer, due to several problems that could manifest. The reason for slow decomposition is the lack of a good environment for soil organisms that perform decomposition. I would guess that the lack of oxygen is the limiting factor. (That's why people turn their compost piles.) The tape and printed pictures with the cardboard is a simple a soil contamination issue, like if you get compost with plastic in it that came from shredding the plastic bags of lawn refuse that made the compost. It looks bad, but really doesn't impact growth much (in moderation, of course).
The capillary action of moisture upward from below as the surface dries, is huge. It's free and
clean water, so you won't be using so much tap water, too. - not so high pH, and no added contaminants like fluorine, chlorine and such.
For that initial weed control, yes, you need to do that if the weeds are perennial. This is a good place to use Round Up, in my opinion. It won't move in the soil, up or down. The Round Up is bound so tightly to soil particles that even roots can't extract it, and certainly not bulb scales. This is the reason why above ground parts absorb Round Up and below ground parts don't: it's not that a particular part of the plant is necessarily more (or less) vulnerable, it's that if RU has the opportunity to bind with other particles first, it will. So if you had clean roots (no soil), they would absorb RU just as easily (actually more easily) than clean leaves. That's why it is stressed to use clean water when mixing RU. Dirty water will reduce its effectiveness because the RU will bind with the suspended dirt, and become, in effect, inert.
Something to keep in mind
always:
Every time you change or do something contrary to the natural, there will be consequences. Look beyond the mere outcome that you perceive as desirable. Sometimes it's all good, sometimes not, but usually it requires a change in management of some other aspect(s).