Would a manuscript be rejected if the cover letter is missing?
It is very unlikely for the paper to be rejected solely for the reason of the missing cover letter.

It is highly likely that you are going to be contacted (using an automated system or personal email) about an incomplete submission or the submission process itself will continue without the cover letter at all.
I would say, missing a cover letter might have an impact on the following:

The potential delay between the journal realizing that the submission is incomplete, communicating that to you, and you resubmitting.

The cover letter can help to assign the associate editor/track editor and, to some extent, select the reviewers. If the cover letter is not present in the submission, it might either take slightly longer or lead to some time/choice inefficiencies that could potentially be avoided.

Sometimes, the cover letter contains substantial necessary information. Say, there was a previous submission to this or another journal some time ago, and you want to inform the editor about that and point out some crucial details that are new in this submission. Or explain why the presence of additional/supplementary materials for your paper is necessary for the review.
In that case, leaving out the cover letter simply does not help to build your case in the best possible way.