That's true, and if you're doing intra-Schengen travel to attend a relative's funeral, I'd consider risking it...
But otherwise: keep in mind that the freedom of movement is reserved to EU citizens (and their families): if you had to get a residence permit (and thus you're a non-EU national), you don't have the same rights.
Even without temporary (Covid, terrorism, etc.) internal checks, you can be stopped by police (national laws will differ) and be asked to show proof of id, and proof of your right to stay.
The fact that you have a pending residence application in progress will usually give the right to stay in the country that you applied in, but that proof might be as flimsy as as a stamped slip of paper (not even a A4 paper with a letterhead) and/or an email in the local language. Don't expect German police to be able to read and accept your Italian piece of paper, or viceversa Italian police to read and accept your piece of paper in German.
In fact, the same applies for non-EU family of EU citizens: the residence permits will be denied only in extreme cases (e.g. terrorism)... if you're just a non-EU citizen, there are even more situations in which that would apply. Imagine that the country that you applied in might refuse your residence permit: you'd then have to leave the country and Schengen (or file some kind of appeal), and that would make it even clearer that you wouldn't have right to stay in another Schengen country.
So, you might not have right to stay (in another Schengen country), and even if you might have it, proving your right might not be easy.
About the
> Please add a link
The original request could've made sense (you could have for example linked to directive 2004/38/EC , but that doesn't apply to people who aren't EU citizens or family of EU citizens)... but note that in this case we're trying to prove a negative (the laws will usually describe which rights you have, but not in which case you don't have such a right... you might find a guidance or case law document, but those are scarcer)
Again: the whole situation is really unfortunate, because the laws are also written with the expectation that you won't have to wait long after applying for the necessary documents. And even when the laws are clear about the timelines, the bureaucracy will try to weasel themselves out of it, for example Article 10 of the aforementioned directive states:
> The right of residence of family members of a Union citizen who are not nationals of a Member
State shall be evidenced by the issuing of a document called "Residence card of a family member of a Union citizen" no later than six months from the date on which they submit the application. A certificate of application for the residence card shall be issued immediately
Of course, when we had to deal with it, the bureaucracy just asserted that until you show up for the appointment (which you had to wait more than 3 months for, since you originally applied), you haven't actually "submitted your application", yet.
As an EU citizen, realizing first-hand how slow, uncertain, and oppressive our immigration system is, really left a bad taste in my mouth... If you're not a citizen of the EU or an “Annex II” country, I wish you good luck when applying for a Schengen visa, but in case that you might get rejected: don't sweat it, and just consider other destinations, if you're planning a vacation... there are a bunch more places with friendlier visa policies (e.g. Turkiye, Cape Verde, Morocco, UAE, etc.)