First off, I love this and I want Pet thresholds to be added, thank you @Izay for bringing it up here.
In your example you mention Giant mole with a drop rate of 1/3000 and a threshold of 1000. Although in RS3 the thresholds equal base drop rate/5 (or *3 for Dagannoth kings as all three share the same killcount) which means once you reach 600 kc for Giant mole your drop rate will be 2/3000 or 1/1500.
I kinda like to compare the games and generally, monsters or bosses take longer to kill on OSRS than they do on RS3, plus the base drop rates of any pets on OSRS are already higher than they are on RS3 (e.g. DKs 5000 vs 2500, Corp 5000 vs 2500, KQ 3000 vs 2500 and the list goes on, these all as well take considerably longer to kill on OSRS than RS3).
Keeping the above in mind I would go with the exact same calculations of pet base drop rate/5 = 1 threshold.
Of course on average this will end up with pets being received earlier than before, but by not altering the base drop rate before the first threshold only the people that actually grind for pets will benefit from this and I think this is a good thing. You will not end up with 6000 Daggannoth Rex killcount and stand there emptyhanded as this kc in fact means nothing.
Up to this day only 6 people would have reached the first threshold for the Alchemical Hydra pet, and even then you need to consider that the drop rate increases but at no point in time is the pet guaranteed to drop.
Again, I love this suggestion and I hope it will be added this way, as it acts as a long term motivation and in fact does not devalue pets at all, as most pets we see now are likely obtained at a very low killcount anyway which would not have made a difference using this system.