Let me try and reason this out. First, if you are someone who is looking at their phd as a 'stamp', there are two extreme outcomes - the best brands and the regular ones. Now, if your intention is to get a stamp, a pedigree and a symbol you should obviously try for a good university. That also automatically checks many boxes. A good branded university usually will give you exposure, research grants, collaborative exercises like publications etc. It will also (possibly) give you a good guide who will teach you stuff. I have seen people who average publications but reaching great places because of their phd. A lot of top profs at NLS and NALSAR have regular publications in peer-reviewed journals but their phd is from the best univs in the world.

Second, you are looking to genuinely make something out of your phd. This again is subject to multiple permutations and combinations. More often than not your phd applications are strengthened by the fact that you already have some publications. So, I am assuming that you wouldn't need help with publications as such. Here, if you are already publishing, care or scopus, I think the univ won't matter. I know people with decent phd's but better publication record.

Third, if you don't already have publications and want your phd to do things for it. I don't think it will work this way. Here, if you have enrolled in a bad univ, you won't get enough opportunities and if you have enrolled in a brand, people will get to gradually know you for your research granted that you work towards it.

From the perspective of recruitment nothing concrete can be said. On pen and paper, if you have someone from Harvard with a phd and no publications, it might seem more of an achievement than someone from a TLC phd with 3 Care Listed ones. The game suddenly will change and perhaps be levelled if the second person has 3 scopus.

Crap! I wrote a lot. Sorry.