A/The town where I was born

[Snappy]

In the town where I was born
Lived a man who sailed to sea (from "Yellow Submarine," a Beatle song written by Lennon and McCartney)

In my understanding, "the" before "town" in the above lyrics is required because it is a specific place restricted by the phrase "where I was born."

I wonder if it is possible to use "a" before the name of a place. Are the following sentences with restrictive phrases possible?

1. There was a famous shrine in a town where I was born. (NOT ...in a town, where I was born.)
2. I took a little trip to a small village where I was born and raised up. (NOT ...a small village, where I was born and raised up.)

[Ann1977]

I wonder if it is possible to use "a" before the name of a place. Are the following sentences with restrictive phrases possible?

1. There was a famous shrine in a town where I was born. (NOT ...in a town, where I was born.)

This doesn't work. The word "a" clashes too much with "where I was born."

2. I took a little trip to a small village where I was born and raised up. (NOT ...a small village, where I was born and raised up.)

I think this might get a pass. The reader might think you are having a little linguistic game --

"I took a little trip to a small village -- namely, the little village where I was born and raised"
-or-
"I took a little trip to a small village that just so happened to be the one where I was born and raised."

[Raymott]

2. I took a little trip to a small village where I was born and raised up.

No, you need "the".
Besides, you can't use a comma when speaking.
Perhaps:
2. I took a little trip to a small village where I had been born and raised up.

In any case, you'd need a good reason to use "a".