What was the ticking clock as regards the burial?
The shuttle had lost fuel, but there was no dialogue to the effect that the leak would persist and more fuel would be wasted by the minute. The crew lacked supplies, but there was no immediate danger of starvation. The starship in the vicinity would be forced to depart in a couple of days, after which she would be gone for a maximum of six days, and then return to continue the rescue efforts. And the natives could be kept at bay indefinitely as long as our heroes did not loiter far away from each other and from the immense firepower they possessed.
Of course they had limited time. The Enterprise was leaving soon. The "immense firepower" was being drained for fuel, so there would be no defense against the creatures (who were out for blood). Search parties were being attacked and killed almost as soon as they arrived. The shuttle crew couldn't continue to use the batteries to electrify the ship because they needed that power. Once draining the phasers, they would only have enough fuel to reach orbit for a short time before being forced to land or burn up. So, yes, they had to get into orbit BEFORE the Enterprise left. That was a major plot point, actually.
It's all right here:
SPOCK: Yeoman, your phaser.
MEARS: But what if the creatures attack again?
SPOCK: They won't attack for at least several hours. By then, with luck, we'll be gone.
SCOTT: If I can get a full load, we should be able to achieve orbit with all hands. Not that we can maintain it long.
SPOCK: We don't have to maintain it very long, Mister Scott. In less than twenty four hours, the Enterprise will be forced to abandon its search in order to make a rendezvous. If we can't maintain orbit after that time, it won't make any difference. If we burn up in a decaying orbit or die here on the planet's surface, we shall surely die.
Regarding the services:
BOMA: Mister Spock. we're ready.
SPOCK: For what?
BOMA: The services for Latimer.
SPOCK: Mister Boma, we're working against time.
Spock did find the time to have a quick burial in the end, but only when it was deemed practical and they were ready to go. Also beacue the creatures were quiet at the time. It never should have been the priority Boma made it out to be.