Ok, I am gonna be the bad guy here

... and I really don't mean to be but there is something here that should be said especially if you are to be goat owners.
neighbor dog began to play a little too rough
1. The grandmother is NOT responsible for anything. PERIOD.
YOU own the kid and should know that it is a herd animal and prey. A dog may chase and "play" causing injury. Albeit a NEWBIE mistake, these are not two animals that should have been together or "played" together. YOUR mistake, not grandmas. You had a responsibility to keep the kid goat safe, that failed and now there are repercussions. The dog is doing what dogs do. Livestock guardians are truly the only dogs that should be with livestock... they were bred and wired to tend to and care for goats/sheep. Pet dogs are a no no and one of the biggest threats to goats.
Being furious that she didn't help my poor baby,
What more did you want the vet to do at this point?
2. You are blaming a vet. If the puncture started out small, as I assume it would, and THAT particular puncture went unnoticed that would NOT be unusual. You were given a shot to administer for 5 days... that more than likely was an antibiotic. At some point you should have noticed either the smell, or the growing in size of the abscess.
How do you miss a softball size wound?
I had missed nearly a softball sized wound on his back.
By now, It had been nearly two weeks since the accident.
3. Once the abscess was noticed you very well could have called the first vet. Instead you blame the vet as somehow it was their fault.
Here is where I am the bad guy.
You should have been checking the progress of the healing everyday several times a day. The fact you did not notice til it was huge, and rotting and had to be debreded tells me you were not monitoring your kid goat.
The fact you did not see the dog do this says you did not have the goat or dog secured.
It is your responsibility to learn and understand what it means to be a goat keeper. Find a mentor and learn what you can.
I share my own recent experience to show you how a vet relationship works.
My Nubian had a small lump on her jaw/mouth area. We know it isn't CL, but were concerned it may be salivary, or tooth abscess.
We take goat to vet, vet examines, also gets bit in the process... not quite convinced it was either but it was obvious a small lump was there. DID NOT feel it was best to lance at that point but to run a course of antibiotics and see if it clears up.
We ran the course but to no avail it did not decrease but increased in size.
This is where we come in- OBSERVATION and communicating with the vet. OUR responsibility.
Took goat in again because vet decided it would be a good idea to x-ray... making sure it isn't a bone issue...
We did x-rays, abscess was growing larger at this point.
The abscess was lanced and drained - NASTY! Large lance and drained out, put on antibiotics again til results came back from culture. Directions were to clean with betadine 3x day.
Same day-
5 pm goat was fine but a little swollen. Late that night by 10- 10:30 pm Her face was completely swollen.
Vet made emergency trip next day. The lance had clotted and nothing was draining... she was given a mild sedative, lanced open again and this time another hole to put a drain in.
In the end we and the vet think a splinter entered from the outside of her face and just got deep down in there and brought the whole thing about.
I share this because there are often steps in a situation... observation, keeping a close eye on a goat, monitoring are all husbandry skills. The goat owners involvement is CRITICAL in the care of their goat. Working with a vet and having and building a partnership is very important.
So was it my vets fault the lance had clotted and was not draining?
NO- What if we weren't checking her regularly?
Should I blame the vet that the original course of treatment didn't clear up the lump?
NO
I see no fault on the original clinics part I do see failure to be attentive and care for your goat kid that brought about the original problem (bite) but also contributed to the worsening condition (abscess/infection).
Suck it up, and take responsibility... this was not your vet's fault. Not once in your post did you mention anything you could have done differently... no, I didn't realize how dangerous it could be for a dog to play with my goat... no- I should have been putting my hands on him everyday and checking to see how the wound was healing ...
As a goat owner I am aware of how fast an abscess can form and I understand you are new and may not realize that these things can happen and happen fast. Experience will make you more aware.
And since no one has mentioned this your goat needs a buddy. They are herd animals and need another goat.
This may seem a bit harsh and I really don't want it to come off that way.. what my hope is that you will become a great goat owner... I am very glad to hear he is on the mend now. Is he wethered?