No ointment or cream is going to prevent pus from getting out and the oxygenation needed comes from the blood flow not external exposure.
Yes first and foremost above all else is copious irrigation. NOT with iodine or alcohol which definitely causes more harm by way of tissue necrosis than good by reducing bacteria counts.
Healing is then best facilitated by maintaining a moist environment. Best is a hydrogel either applied or as a dressing. Moist keeps the tissues from drying out and allows all the body’s healing messengers and cells (including cytokines and new skin cells) to get where they need to be fastest.
For non surgical wounds a topical antibiotic ointment does help reduce local infection risk in addition to keeping the wound moist. Petrolatum fine for surgical ones, and it helps getting the stitches out easier too as it cuts down on scabs drying over them.
That said most minor home wounds will do fine pretty much no matter what. I don’t personally bother to stock a hydrogel and usually just wash off well, pressure until bleeding definitely stopped, and slap on a bandaid. But it is not best.