"That when the President of the US needs a break in the price of gas, like around elections, he begs the Saudis to pump more oil?

And the Saudis are then touted as our allies because they do it?"

It has more to do with a display of solidarity.

It could also be that a visit that would normally would go mostly unnoticed [and not really has much to do with oil supply] gets more press and questions asked by reporters has to do with oil supply.

The Saudis are the largest exporter of oil in the world. The Saudis do have a close and almost unique relationship with the US. This special relationship adds to Saudi standing among other middle eastern nations {the US is the superpower and has a lot global influence}. And though being the largest oil exporter in the world has a lot influence and a means of generating wealth; for Saudi Arabia the more important aspect of this country is it's the location of Mecca.
E.g. "Every year some 2 million pilgrims attend the hajj"
http://i-cias.com/e.o/mecca.htm
And of course, every Muslim in the world prays facing towards Mecca.

So, the Saudi rulers see themselves and others see them as very significant leaders of entire Middle eastern region and this way, do not just represent Saudi Arabia, itself, but also other major arab oil producers.

But Saudis are basically pumping oil as fast as they can and at fairly constant rate. It's a train going 50 mph- if they slow the train they lose money, though they may be able to speed up the train to 55 mph.
Look, the US almost produces as much oil as Saudi Arabia and I would say Saudi Arabia [mainly because of it's infrastructural capacity] can change it's output rate a bit more easily, but essentially it's similar degree of control as US domestic oil production.
What are bigger factors in terms of price is the market's perception of future oil supply. And if market pushes the price up, then producers may decide to invest more money to increase production [which may take months to years to bring about this increase in production] meanwhile market prices can jump up or down $10 a barrel in an afternoon.