Misshandling of Postgres Pools

[salmosri]

postgres-mcp doesn't properly manage database connection pools when using SSE (Server-Sent Events) transport.

Here's what happens:

1. New pool per connection: Each time an MCP client connects via SSE, postgres-mcp creates a new database connection pool (pool-1, pool-2, pool-47, etc.)
2. Pools get closed: After handling a request, the pool is closed but the reference remains
3. Stale pool references: Subsequent requests try to use these closed pools, resulting in the error "the pool 'pool-47' is closed"
4. No pool reuse: Instead of reusing a single connection pool across all SSE sessions, it creates new ones and doesn't clean up properly

This is a design flaw in how postgres-mcp handles SSE connections. The SSE transport is meant for long-lived connections, but postgres-mcp treats each request as isolated, creating this pool management issue.

Evidence from the logs:
ERROR Error executing query: Connection server.py:401
attempt failed: the pool 'pool-49' is
closed
ERROR Error executing query: Connection server.py:401
attempt failed: the pool 'pool-55' is
closed

The incrementing pool numbers (pool-49, pool-55, pool-67) show that new pools keep being created instead of reusing existing ones.