I have a set of new LED bulbs which want hot & neutral connection at just one end of each bulb.
The fixture in question however is wired differently than I expected. It has just one wire per end. The ballast diagram matches the fixture wiring:
Examining the bases they indeed have just one electrical connection per side. What can be done to bypass the ballast in this situation? Trashing the entire fixture is an option (and may be less trouble than messing with finding compatible bases). No new bases were included with the lamps, nor would generic bases fit due to shape of the fixture.
Those sockets at the end of the fixture that the tube snaps into are called lampholders.
You are correct that fluorescent tubes have 2 pins on each end (a preheat filament is between them) and some ballasts do not use that feature. To avoid having to make a bunch of jumper wires, manufacturers use a shorting lampholder which internally shorts both pins. You are cursed with those.
Virtually all lampholders come down to 3-4 different compatible styles (overwhelmingly one**), all of which I've been able to find online. I've never had to abandon a fixture due to odd lampholders. That seems even less likely with a shorting lampholder. Get some non-shorting lampholders. Done.
In the future you avoid this problem either by making sure you are buying direct-wire LED "tubes" that feed from opposite ends (that's safer anyway) or buy those particular plug-n-play LEDs made for your ballast type (but then you have to maintain the ballast, no thanks.)
** I just overhauled a prewar fixture. The 4' fixture was bulletproof drawn 18 gauge steel and weighed 10 pounds without the ballast. The lampholders were trapped in a weird way, but guess what, they were the most common type. I put a T8 ballast in it, I go for the 90CRI real tubes, better light quality.
The LED lamps you bought do not require the ballast you have in your photo.
Remove the ballast.
You will need to change the pin holders (tombstones). The lamps I bought came with new ones. But if these didn't you will need to buy some.
Then take 120 volts, the black (hot) and the white (neutral) wire, directly to the power end of the LED lamp.
Good luck and be careful!