I'm sorry for such a large answer, OP, I've written just a part of what I can say on this topic, but it is already huge to read and if I just say the wayshrines were switched off and destroyed, my opinion below would not be understood properly for sure. Your question is the one I asked myself too among many others I asked myself on this topic several years ago. So here I share my thoughts.

I agree with all the fellow players above and especially with @RaddlemanNumber7 here. Those wayshrines of the eras ahead are not the same in comparison to those we see today in 2E 582. There are two major differences: the first difference is their origin - the wayshrines in Dawnguard were constructed by the Falmer and the ones in Cyrodiil we see in the 3rd Era were built by the Ayleids. None of the types is used today in ESO. We use the modern Transitus Network maintained by our alliance mages to travel between the keeps in Cyrodiil and the only real modern wayshrines situated there are the ones at the gates to travel back to our home territories. The second difference is the wayshrine's design - even if we will be able to use those Falmeri wayshrines it's braziers should be lit just like all the other modern wayshrines we use in order to make a wayshrine operative. Do you remember any of the 3rd and 4rth Era's wayshrines with their braziers lit? My memory tells me there are none of them there.

Well, the point of my post is the following: since we have no lore developed regarding the fate of all the Soul Shriven yet, and considering the wayshrine difference I've mentioned above, my logic leads me to thinking of the only logical answer to your question I find possible today, OP: to permanently destroy or banish all of the likes of us the challenger must strip them of all the soul gems they possess or somehow to change their very nature - say, to sort them out into the White and Black soul gem types, to change the soultrapping spell and ban the one used today . Ok, then he must kill some of us twice in order to make the Soul Summons passive go off. And finally the challenger has to destroy or switch off the braziers of all the wayshrines in Mundus or at least on Nirn. I emphasise it - this is the only way a mortal denizen of Mundus can stand against us.

A hard task, isn't it? So, for the change of soul gems and the soul trap spells we have to "thank" our old friend Vanus Galerion for he has been planning it for quite a time already. And as we know it, his plans will come true in about 700 years. To destroy or switch off all the wayshrines a challenger must face a great threat to his life or authority we pose to him. Thus he must either ally with other challengers all over Tamriel or simply conquer Tamriel, so nobody opposes his will. The best known candidate to fit this role is Tiber Septim. Well, at least yet, of course. He conquered all of Tamriel, he might have used Numidium as the only possible mean to overwhelm us, and since that time he was also able to destroy or turn off the wayshrines everywhere in Tamriel and the Mage's Guild with it's Mages Guild monopoly on all public magical services given by one of the terms of the The Treaty of the Armistice, such as Soul Gem policy, spellcrafting, teleport service and scrolls of divine intervention became quite a good tool to do that in every part of his domain.

Of course, this is just a theory based on my logic of the existing lore, games mechanics and the lack of certain answers in the lore all combined. It all could have happened before Tiber Septim's times due to some lore phenomenon yet not created. But if the loremasters won't subject this part of the lore to the glare of truth, then it is highly likely to be the way I've written above. Thanks for reading and have a nice day !