Use a connection pool instead of a single DB connection (#408)

<!-- Provide a brief summary of your changes -->

## Motivation and Context
<!-- Why is this change needed? What problem does it solve? -->
The following PR should fix the recent "conn busy" issue caused by the
PostgreSQL implementation using a single database connection (*pgx.Conn)
instead of a connection pool (*pgxpool.Pool). When multiple concurrent
requests try to access the endpoint, they would conflict trying to use
the same connection.

## How Has This Been Tested?
<!-- Have you tested this in a real application? Which scenarios were
tested? -->
Builds locally

## Breaking Changes
<!-- Will users need to update their code or configurations? -->
No

## Types of changes
<!-- What types of changes does your code introduce? Put an `x` in all
the boxes that apply: -->
- [x] Bug fix (non-breaking change which fixes an issue)
- [ ] New feature (non-breaking change which adds functionality)
- [ ] Breaking change (fix or feature that would cause existing
functionality to change)
- [ ] Documentation update

## Checklist
<!-- Go over all the following points, and put an `x` in all the boxes
that apply. -->
- [ ] I have read the [MCP
Documentation](https://modelcontextprotocol.io)
- [ ] My code follows the repository's style guidelines
- [ ] New and existing tests pass locally
- [ ] I have added appropriate error handling
- [ ] I have added or updated documentation as needed

## Additional context
<!-- Add any other context, implementation notes, or design decisions
-->
Related to #400, #405 and #407

---------

Signed-off-by: Radoslav Dimitrov <radoslav@stacklok.com>

go.mod

@@ -32,6 +32,7 @@ require (
 	github.com/grafana/regexp v0.0.0-20240518133315-a468a5bfb3bc // indirect
 	github.com/jackc/pgpassfile v1.0.0 // indirect
 	github.com/jackc/pgservicefile v0.0.0-20240606120523-5a60cdf6a761 // indirect
+	github.com/jackc/puddle/v2 v2.2.2 // indirect
 	github.com/munnerz/goautoneg v0.0.0-20191010083416-a7dc8b61c822 // indirect
 	github.com/pmezard/go-difflib v1.0.0 // indirect
 	github.com/prometheus/client_model v0.6.2 // indirect
@@ -41,6 +42,7 @@ require (
 	go.opentelemetry.io/auto/sdk v1.1.0 // indirect
 	go.opentelemetry.io/otel/trace v1.37.0 // indirect
 	golang.org/x/crypto v0.41.0 // indirect
+	golang.org/x/sync v0.16.0 // indirect
 	golang.org/x/sys v0.35.0 // indirect
 	golang.org/x/text v0.28.0 // indirect
 	google.golang.org/protobuf v1.36.6 // indirect