Assuming you have no firearms or archery equipment, how would you harvest birds in a survival situation beyond trying the old pull-string-to-jerk-stick-supporting-container trick?

It depends on a few things. So a few methods.
Note that almost all of these would be considered poaching under any conditions except dire necessity.
1: Snares, there are several types that can be set. Most survival texts give 2 types.

2: Box or deadfall traps set with triggers like the Figure 4 or the Paiute

3: Walk in funnel and ladder traps. It is possible to create traps which birds can enter but are unable to figure out how to get out of. Usually they are baited mazes with the tops closed.
The ladder trap is an old trick for catching crows or magpies.
The funnel trap is even easier to make. The entry could be a vertical offset V instead of a funnel too.
They show some on this page.
http://icwdm.org/handbook/birds/Magpies.asp
It seems they would be a lot of work to weave out of sticks (wicker?) but I suppose if you had nothing better to spend your time on it would at least keep your mind occupied.

4: Nets, if you have netting you can set it as a trap, as a net you can chase birds into and if you are in a place the birds fly through as a net to tangle them out of the air. Sometimes you can set up netting and herd or bait a flock of birds underneath it.

5: Run them down, if it is molting season the birds can't fly and you might be able to chase them down. Of course with ducks and geese you have to cut them off from the water because they can easily out swim you.

6: Hit them with a rock. Kids in the north often hunt spruce hens and ptarmigan with sticks and stones. (how good is your aim?)

7: Pluck them off low roosts at night. Some birds like spruce hens roost up in trees but usually not very high and sometimes let you get within reach. You might want to make a capture noose with some string and a pole.

8: Raid nests. You are not actually getting the bird but eggs and nestlings. If you are in a hatchery or someplace where a lot of birds nest this can be very effective.

9: Use a bolo. This works for low flying birds. The idea is to tangle them up so they fall to the ground. It takes a bit of practice to know how to use one.
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Bird lime has been used as a method of catching small birds, but I don't think it would pay to spend the time to make the bird lime. Usually it is made by boiling linseed oil down to very tacky goo or fermenting and boiling down holly bark and mixing it with oil.

Rabbit stick, or bola.

sling shoot

sling shot, rabbit stick.

Although, I am hoping that I learn something by watching "Men who stare at goats" with George Clooney and Ewan McGregor next month.

Crocodile Dundee could stare down and water buffalo so it could work.

but seriously, I train with a slingshot in my backyard.

check out page 264 of Richard Graves "bushcraft".it show's how to make a bird snare by putting many small loops of fine line,he said horse hair,on a stick and putting it in a tree.it's so effective that it's illegal.

Slingshot gets my vote.

Many birds eat fruit. Others (like crows) are scavengers, you could use leftover guts. Set your fishhook (with line anchored to something) in that.

Or sticky traps.

Sue

You could use an airgun. Pretty quiet, can be very accurate, and the ammo is cheap.

Does your BOB include makings of snares, a slingshot, or an air gun?

don't forget the plain old method of pissing off the bird and going hand-to-beak with it. I'm thinking specifically of Canadian geese, which seem to be pretty territorial. I've been charged a few times, and it seems it'd be easy enough to let it come at me and just break it's neck or choke it out.