Why do lead-acid batteries perform better than Li-ion with substantially more ampere hour?
I can see a few issues:


The UltraFire batteries shown in your photo (and in your calculations) are fake. It is impossible to have 12,800 mAh capacity 26650 batteries. This is similar to my answer here about 18650 batteries. The largest capacity 26650 batteries currently available are less than 6000 mAh.

For example:

UltraFire BRC26650 7800mAh (Red) - test by lygte-info.dk
Claimed capacity 7600 mAh, measured capacity around 2000 mAh, but unable to sustain fast discharge and mismatched capacity between batteries.

UltraFire BRC26650 7300mAh (Gray) - test by lygte-info.dk
Claimed capacity 7300 mAh, measured capacity around 2500 mAh, and mismatched capacity between batteries.

Various larger Li-ion battery capacity test results, including 26650 - lygte-info.dk
Note that not all batteries on this page are 26650, some are 21700 etc. Search for 26650 on the page, if that is the size you are interested in.

Batteries with ridiculous ratings (actually comparisons of good &amp; bad batteries) - lygte-info.dk
You can see some batteries with clearly fake ratings, as well as capacity vs. discharge rate available from different genuine models.
Not all Li-ion / LiPo batteries are equal in their ability to deliver all the energy inside. Currently, with 18650 / 26650 batteries, there are effectively "medium capacity, high discharge rate" battery models, and "high capacity, lower discharge rate" battery models.

If you get the highest capacity batteries, you are unlikely to be able to discharge them as quickly as some lower-capacity batteries. This is not my specialist area, but from the research I did a while ago, there appears to be a trade-off between the battery manufacturing processes for these two features - capacity and power delivery (effectively the internal resistance). This is a topic frequently discussed in flashlight and RC model forums.

You need to investigate the "C" discharge rating of whichever Li-ion / LiPo  batteries you are considering, and read their datasheets very carefully, to understand how quickly you are allowed to discharge them (safely). Just looking at their capacity (in this case, the fake 12,800 mAh) does not tell you how quickly you can discharge them, nor does it tell you their internal resistance (which is related), even if the capacity value was true (which it isn't).
Building your own battery packs, without any form of BMS or other protection (as shown in your photo) is high-risk, due to the lack of balancing or voltage control for the individual cells. That is a whole topic on its own.


(P.S. Soldering directly to 18650 / 26650 batteries does them no good at all, and can easily damage the internal plastic separator sheets, leading to reduced capacity or even overheating, fire and venting. Can people get away with doing that sometimes? Yes. Is there a risk in doing that? Yes, you will see warnings on 18650 / 26650 battery datasheets not to do it. There are good reasons why professional battery pack manufacturers don't do that.)