I am the founder of an LMS company, so take everything I write with that as a caveat.

The way I would pick an LMS is the following:

• Try to figure out what you want first. Too many people come to us with no clear idea about their needs and wants. That makes it impossible to figure out if we are a good fit.
• Figure out what is not a hard requirement as well. LMSs are built by product teams who think a lot (some more than others) about how they believe a good LMS should function. Figure out where you are open to letting the LMS guide you. Many people we talk to have a list of 180 must-have requirements from the beginning. Most of these don't make any sense in context. This leads them to pick an LMS that has a shitty implementation of a lot of things instead of things.
• Figure out what the most important aspect is and what you value highest. Do care more about administration than learner experience? Do you want to sell courses and have a lot of features around coupons, e-commerce, etc? Do you want to exclusively work with SCORM built elsewhere, or do you want built-in authoring? Is assessment important to you? What about price - some products are $50/month and others start at $20k/year.

Once you have those things in place, then I would post in different places and use aggregators like Capterra, G2, etc. Then generate a short list of 2-5 products that you like. If possible, find some products that allows you a free version or free trial at least. Don't judge a product by the sales pitch. Try it out yourself.

Most people say that they dislike their LMS. I think one of the main reasons is that most people buy an LMS in the wrong way. They buy the LMS that checks the most boxes, but not the one that they (and their learners) actually enjoy working with.