The way the 3DS hardware works is that it indeed does have a DS mode, but one that is completely disconnected from the 3DS hardware features, meaning any DS game launched on a 3DS cannot access its hardware features such as the SD card or internal storage, which both do not exist during DS mode.
Additionally, the 3DS hardware (during 3DS mode) cannot even connect with another DS, unlike the DS which had a second slot and hardware that allowed it to utilize GBA cartridges. A bit of food for thought, this connectivity is what allowed hackers to make the first flashcards for the DS, which were exploits that ran through Slot Two mode.
But back to the subject at hand, the only way possible to allow a 5th gen game to connect to another game available on 3DS is to exploit their online connectivity and their ability to access cloud storage (possibly in the same way Dream World is accessed) and then have the 3DS game connect to that same cloud and get the data back.
To put it more basically, the only possible way to have Gen 5, a DS game, connect to Gen 6, a 3DS game, is through using Wifi, which in turn require separate servers, which need money to run.
Now my point here isn't to try and defend TPCi and Nintendo for their pricing decisions, that is subjective.
But what I want to do is to explain why they had to resort to to using an online method to allow for cross gen, and if you don't believe me, you can check this tweet:
https://twitter.com/Junichi_Masuda/status/375616372421562368where Masuda confirms the reason for using Bank is due to connecting B/W on DS with X/Y on 3DS being impossible.
And to be honest, cross gen compatibility has always been a hassle save for Gen 4, and if you usually borrowed your friend's DS/GB to transfer pokemon, just borrow their 3DS now and use their Pokemon Bank account which is confirmed to work on multiple copies.
TL;DR Nintendo hardware is really funky to work with when it comes to this stuff and this is their only workaround.
