Have you tried clearing the data cache, deleting unused apps and even doing a fresh install of everything on your Nexus 7 2013?
Can you get micro USB memory sticks (perhaps OTG?) that will make up for the Nexus’ default memory shortfall?
I recently bought a Maze Alpha mobile, it’s really zippy compared to my Nexus 7 2013, but the extra ~2″ of diagonal screen makes web surfing and especially movie streaming much more of a joy on the Nexus!
I only bought the Maze as my first “smart” Android phone as something that could make/receive calls and texts for those times when BT hotspot reception is flaky or even non-existent, using a prepaid Three 12GB data sim card and a separate “3-2-1” sim card for traditional mobile use. Plus I could use the mobile data to make use of the “live tracking” feature on my Lezyne Super GPS, so selected e-mailed people like Mrsgoat can see in real-time where I am while riding.
Despite their con-artist nomenclature, the only “Amazon Fire HD” tablet that actually uses a 1920×1080/1200 display is the new 10 IIRC! They were ~£110 around Black Friday, think they are ~£150 now.