Learn English – Never fail to disappoint

I imagine it's an echo of this expression (as defined by OED)...

(strictly) for the birds
trivial, worthless; appealing only to gullible people. orig. and chiefly U.S. colloq.

Dobbs clearly isn't interested in anything Howard has to say. He's implying that despite Howard's "evangelical" enthusiasm, nobody else would be interested either, except gullible people as above.

OP is on the right track with bird = fellow - but here it's specifically birdbrained (stupid) fellows. The word catch is also used figuratively here (Dobbs means interest, draw in, captivate, impress).

Maybe some people use it to mean "a goal was attained and now they don't know what to do", but in this context Harris isn't saying that. He's saying that a dubious or worthless goal was attained, regardless of what happens next.

If you race your friend to a chain-link fence and you win, only to find out the fence was electrified, you're the dog who caught the car.

For the dog analogy, either it dies or nobody in the car is concerned. The dog's aggressive attempt to achieve "catching" the car is a miscalculation. There is zero intent to convey a goal or achievement was reached, because the goal was to attack the car which is the silliness implied in the statement. The "dog" is looking for a fight it doesn't know it cannot win.

You can infer a "what now?" effect, but the gist is, it's playing out of its league and doesn't know it.