We have a beautiful birch off the deck surrounded by a mulch bed. Over the past 3 years, I have attempted to plant various plants in the surrounding mulch...azaleas, lavender, and zinnias. Despite enriching the soil when I plant, nothing does well in this area. Is the birch changing the acidity of the soil, and would you have suggestions for why nothing seems to grow well in that spot?

Birch Problems
The birch is not changing the soil pH it is just plain out competing the other plants.
Birches have a very fine feeder root system. Gardening around one is next to impossible, especially with annual plantings.
The three you mentioned will be difficult. First zinnias are annuals and you will need even moisture and extra food.
Lavender is just a pain to grow anywhere
Azaleas are also difficult.
My recommendation is a nice layer of mulch and maybe a juniper or some other durable ground cover.
On a personal note, in my garden the mulch looked the best. I removed the birch after an ice storm. It was the best decision I made as the garden looks so much better.
Dennis - Johnson County Extension
KSP you might be fighting
KSP
you might be fighting with the birch's feeder roots which are resource hogs. They are simply out-competing other plants for the nutrients and water.
You can play the game of amending more, watering more and adding more fertilizer but in the end I think you are only feeding the birch.
You might try just planting something underneath more vigorous and less demanding like ivy or pachysandra. You might have to give them more water and some fertility at first but once established they should do fairly well.