A lot of deities have dogs. They is even a prayer to the dogs of Rudra.
- To Rudra's howling dogs, who swallow their food without blessing, who have wide jaws, I have made this obeisance.
- Through the air he flies, looking down upon all beings: with the majesty of the heavenly dog, with that oblation would we pay homage to thee!
- The three kâlakâñga that are fixed upon the sky like gods, all these I have called for help, to render this person exempt from injury.
- In the waters is thy origin, upon the heavens thy home, in the middle of the sea, and upon the earth thy greatness. With the majesty of the heavenly dog, with that oblation would we pay homage to thee!
- The two dogs of Yama, the black and the brindled one, that guard the road (to heaven), that have been despatched, shall not (go after) thee! Come hither, do not long to be away; do not tarry here with thy mind turned to a distance!
I think a much better explanation for the apparent disregard of dogs is that Devas are afraid of them. This can be explained with fated nemeses of the Devas, who are dogs. This is like the four horses of the four horsemen of the apocalypse mentioned in the Bhavishya Purana are to humans. I hope we can realise hating horses or thinking they are impure based on that is not going to accomplish anything good. These verses make more sense in that light. The long-tongued dog is apparently especially troublesome for the Devas.
With dog the Gods, perplexed, have paid oblation, and with
cow's limbs in sundry sacrifices.
- FOR first possession of your juice, for the exhilarating drink,
Drive ye away the dog, my friends, drive ye the long-tongued dog away.
Dogs also seem to be good at being nemises, even if not malevolent to the completely innocent. There is a dog in the Ramayana, who despite not using any weapons, manages took seal a much worse fate for a Brahmana who wronged him. He did this instead of asking for punishment giving him a blessing he knows he will misuse and pay even worse for. Such cold blooded vengence is fitting for the nemises of the Devas. Read about it here.
It is only a few dogs/one dog that the Devas are worried about. The issue is they do not know which ones are the dangerous ones fated to be their mortal enemies, so they just avoid contact with all of them. Devas (with the exception of the paranoid Kali, who stays much further away) are okay so long as they are not in sight of a dog. It is possible that they are like Vishvamitra, Devapi, and Maru and lead normal-ish full lives before they start fully leading down their fated, legendary destiny.
The explanation of how a the dangerous dogs can get so powerful, while remaining dogs, seems to be given in the Satapatha Brahmana (part of Yajur Veda).
It is worded weirdly, so its meaning is kind of unclear, especially without context from later scriptures. Here a dog, probably one of the dangerous ones, interrupts a horse sacrifice. While it is killed, the wording implies it is still a threat. Then it is (temporarily) stopped by a superpowered horse. The dog is also four-eyed, implying it is related to the dogs of Yama.
But his wicked enemy seeks to lay hold of him who performs the horse-sacrifice, and the horse is a thunderbolt;--having killed the four-eyed dog, he--with 'Undone is the man! undone is the dog!'--plunges it under the horse's feet: it is by means of the thunderbolt he thus stamps him down; and the wicked enemy does not lay hold of him.
This suggests one of the main reasons the dangerous dogs are so difficult for the Devas to deal with is they cannot be kept dead and constantly reassemble their bodies as the most insanely powerful zombies ever. At most all the Devas can do is delay them. Plus the dogs probably get stronger over time, due to experience.