I think I have a solution. I was digging into this myself while working on ways to move from SharePoint Online to Dataverse. I'm trying to get things into a model-driven app format. It's not a perfect solution, and likely is not how you imagined your workflows going... But now you have the information to choose your own strategies. I'm going to try work with this is an attempt to keep data structures as close to elegant as I can. Best of luck to us all.
Here's the trick: Your many-to-many lookup is already accessible in your table forms and views(!?) after you have created the many-to-many relationship in your table. You just need to know where to look and how to use it. It can be non-intuitive at first, but it is viewable in the "Related" tab after you have created a new row of data in the table which you wanted the lookup added. Note: without trying to data from the many-to-many or the related tables. In retrospect, it kind of makes sense, because the hidden join table can only contain rows of data that already exists in the joined tables; you need each related table to have unique data before you try selecting items from one for the other.
I'm including a visual guide because that's how I learn best and how I hope to find these solutions 🙂
This guide assumes you have two dataverse tables created, the "lookup" table has data in it, and that the "looking" table (or both) has had a many-to-many relationship created relating the two tables.
Pic 1: Form Editor showing where you will find everything. But you can't open it in form edit mode. Very tricky, Microsoft folk.

Pic 2: New entry view. Step 1: create your new row of data. You cannot see the relationship table/lookups/anything. Don't be discouraged.

Pic 3: Table view of your new data. Step 2: open/edit that data row.

Pic 4: Form view showing your data for editing. Note the "Related" tab. Step 3: Click it, your many-to-many relationship you set up with another table is there!

Pic 5: View of your many-to-many relationships with the other table. Step 4: You can "Add Existing "to pick from your related table. In the pic, I have 2 items from my related table showing that I already added.

I really hope this helps someone. It's not the answer we all want. However, it's an answer that doesn't seem to add extra work or workarounds, and may just be a matter of teaching end users how to use the tools. Of course, we can also ask Microsoft to build that many-to-many lookup we all dream of.