This card is Intel Wireless 7265D, you can find out by searching through demsg and looking for:
Detected Intel(R) Dual Band Wireless AC 7265, REV=0x210
This card is limited to 2.4 Ghz AP mode, while supporting 5 Ghz client mode. Yet 5 Ghz is certainly possible, something is preventing the card from being able to emit at 5 Ghz.
You can prove this by using:
iw list
And checking if ANY of the 5 Ghz frequencies doesn't have a no IR flag. If at least one doesn't have it, this means the card can emit 5 Ghz, and it means you can configure hostapd to set it to 5 Ghz access point.
However currently the card doesn't allow it. This is also mentioned in the iwlwifi documentation: https://wireless.wiki.kernel.org/en/users/drivers/iwlwifi.
Now what's worse is if there's a disabled flag, if there is, then it's even worse as the hardware doesn't support it. But here with no IR, it's not that the hardware can do it, it's some regulatory setting preventing it from supporting 5 Ghz emission.
Various mailing list archives indicates people just say that the operation is just not supported. Also mentioned is in Intel Support, and the customer service just says that it was not designed to be a 5 Ghz AP: https://communities.intel.com/thread/60677
I suspect that there is a hardware/firmware check for the regulatory domain, which is by default set to World/Global, and this prevents 5 Ghz AP mode. This appears to affect all the Intel Wireless Cards all the way to more recent 8265.
This thread http://forum.ipfire.org/viewtopic.php?t=16231, indicates that the Intel 00-World setting is a hardware/firmware lock. The card needs to be reprogrammed to allow it to set regulatory domain. Does this mean reflashing the firmware of the card?
Ultimately this post shows us we should just get Qualcomm Atheros Wireless cards and avoid Intel or other cards.
It appears that Killer Wireless are Qualcomm Atheros Wireless Cards.
https://vincent.bernat.im/en/blog/2014-intel-7260-access-point
It's important to watch out for systemctl status hostapd and view the errors and enable the important things.
Before buying any hardware make sure to take a look at: https://wireless.wiki.kernel.org/en/users/drivers