I've been using Publisher for a while and love it how light it is comparing to similar competitors and how much heavy load it can handle.
But I was expecting some other updates such as ability to copy paste pages between two open project as well exporting to interactive fillable forms pdf files... well, hopefully something like that comes in before V3.
If you had a single Designer file with 1 Artboard per page it would be easiest. You could simply Open the file, and let Publisher convert the Artboards to pages.
With multiple files, I think you can only copy/paste them one at a time onto Publisher pages if you want to get the content directly into Publisher, or Place them one at a time. But there is a way to make the Placing simpler.
For 160 Designer files, create a Publisher document with 160 pages and a Master Page.
On the Master Page, create a Picture Frame with the size and placement that you want. That Picture Frame will then be on all your document pages.
Select the Place Image Tool. On the file selection dialog, select (for example) the first 10 Designer files. They will appear in the Place Image studio panel, wth the first image selected.
Click the tool in the Picture Frame on page 1. The first image will be applied to that Picture Frame, and the second image will be selected in the Place Image panel.
Move to page 2 (click the page navigator in the lower left of the window).
Click the tool in the Picture Frame on page 2.
Repeat until you've used all those images. Then go back to step 3, and choose the next set of images. I would do them in small groups, 5-10.