Is it true that the type of ML model used is irrelevant? I am training a model on a dataset and all types of relevant algorithms I have used converge close to the same accuracy score, meaning that no one is significantly better performing than the other. For example, if you're training a random forest and a neural network on MNIST, you'll observe an accuracy score of around 98%. Why is this the case, that bottlenecks in performance seem to be dictated by input data rather than the choice of the algorithm?

The idea that "ml model doesn't matter" is brushing a few concepts under the rug.
First and most to the point, there's a fundamental question of how well can you predict your outcome given your features. If your income is "income" and the only feature you have is "subject's height", it doesn't matter what model you use, you just can't predict the outcome well. If you have a high resolution picture and the variable you want to predict is "is the main subject of this picture a cat?", this can be known with very high accuracy, however the relation between inputs and outputs is extremely complex. In the second case, ML model definitely matters.
The idea that "ML model doesn't matter" is an observation that in many (but definitely not all) real world applications of ML, the best way to improve predictions is to get more relevant data, and there's only a minimal amount of improvement you can get from switching the model you are using. So if you have an application in which your ML model is not performing up to requirements, your time is often better spent trying to find more relevant features rather than comparing outcomes of 30 different model choices.
EDIT: It's also worth noting that in your example with MNIST, 98% is actually not very good. Convolutional NN have an error rate of ~0.25%, meaning 1/8 as many errors the Random Forrests you tried. While 98% may seem "good enough", note that for certain applications, such as recognizing whether an object is moving or not for autonomous driving, using a model with a 98% accuracy could lead to hundreds of thousands of deaths if uses by millions of drivers. So the difference between 98% accurate, 99%, 99.9%, etc are all very meaningful differences.
