What does Saudi Arabia have to lose by increasing their production (if they can)?
As was demonstrated by the 2020 Russia–Saudi Arabia oil price war, a signficant production increase would lead to a market crash, and require significant corrections by all producers to stabilize the market again. Such move would sour relations wth other oil producers, and cut into Saudi Arabia's own budget; note that during 2020 crisis, USA reportedly took part in pressuring Saudites into cutting their production.
Thus, at this moment, Saudi Arabia does not want to get into another price war and also finds unlikely that USA would react with some kind of sanctions to their refusal to increase exports.
Selling oil at higher price
One possible reason is that Saudi Arabia might benefit more from higher oil prices - the demand for oil is pretty inelastic, in the sense that the price would have to go really high for many people/industries to start using less of it. Thus, Saudi Arabia simply can sell the same amount of oil at higher price (Oil market is not a genuine market, in the sense that there are few sellers and they typically coordinate among themselves, so there is nearly no competition that could drive down the price.)
Opposing the Iran deal
Another aspect is the Biden Administration attempts at reviving the nuclear deal with Iran, concluded by Obama administration but abandoned by Trump. These efforts were intensified with the beginning of the Russian operation in Ukraine, as Iran could potentially provide great quantity of oil as well (not surprizingly Russia tried to use its influence on Iran to delay the deal).
Already in 2015 the Gulf States and Israel considered the deal as a sell-off, virtually guaranteeing that Iran would acquire nuclear weapons. Their concerns were then offset by promises of extensive arms sales and military cooperation. Unsurprizingly, these countries have adopted an even harder stance now, in view of the progress in the nuclear arms development that Iran has made since the deal was abandoned by the US, with the latter again trying to reassure them by boosting their ability to defend themselves against an Iranian attack, e.g., see here.
After this gesture Saudi Arabia (and it seems also the other Gulf countries) is taking the blame for the inaction on the oil markets. But this is not the only case where the US are not very eager to follow up nor to push as the usually do.
So, Saudi Arabia did nothing because Biden request was just a posture.