Agreed. This is not a robust resilient alternative to GPS. As you point out, it cannot be unless they re-work how cell stations do frequency control.
It's a non-trivial problem though. Whilst one could fit the base stations with high quality low phase noise reference oscillators, that in itself is not a long term solution. It would tide the cell network over short GPS outages, but after a while the cells would have diverged from an agreed timing and the network stops operating. They need that common external reference to keep the whole network in sync, which is why they picked up on GPS in the first place. Using the cell network as a back up to GPS positioning doesn't work, because anything likely to take down GPS in a serious way is going to mean no GPS services for years. There's no way that even the finest low phase noise oscillators in base stations could keep the networks in sync for that time.
There are alternatives. Radio clocks - e.g. Rugby / MSF. Run off the UK's atomic clock resources, it's always right, but I don't know if one can sync an oscillator to it sufficiently accurately for the purposes of a cell network; the signalling method for radio clocks is something like 60kHz, and a very accurate 60kHz, but the time of day at the receiver is good to only 1millisecond (GPS receivers can do far better).
Other technologies are also likely limited in a similar way, e.g. eLoran. That too can carry a timing signal, but is also a low frequency narrow bandwidth signal (which is what limits the timing resolution that can be achieved). That's not a reason to not (re)build eLoran, as that in itself offers a pretty good location service. It'd be far better to put resource into (re)building the eLoran transmitter networks than it would into bodging up a location receiver using cell networks.
Better Long Term Solution
Probably what they will have to do is to redesign the cell networks so that they can distribute time themselves. It would require all the cell stations to be able to hear at least one other cell station in the network, but if that were arranged then they could build a self synchronising network. That fits a number of things nicely
First, there's pressure from the UK gov for companies to share cell sites, stop the mad dash for prime sites amongst the industry players. The ultimate conclusion is that there's one single physical cell network, with all the current "operators" becoming virtual networks on it just as the smaller providers like Virgin and Giff Gaff are today. Merged into one physical network, it's more likely that all the cell base stations are within reach of at least one other cell base station.
Secondly, resiliently self-synced like this, then the cell network does indeed become a solid, reliable, multiply redundant source of time and position services. In fact, they could re-engineer the lower layers of the 5G stack specifically to provide time and position services, rather than such services having to be synthesised (badly) from cell emissions as signals of opportunity.
Won't Happen unless kicked
Thing is, this won't happen spontaneously. A self syncing network is going to cost operators money in one way or other, either through the use of bandwidth for synchronisation, added base stations to provide the synchronisation grid, etc. To make this happen, several major governments around the world are going to have to intervene in the market and pass laws requiring that such things are required by the licensing regulations for cell networks. That's going to take political cooperation between some pretty major governments currently holding adversarial positions against one another...
The Western world could go it alone in this regard. It's going to cost money, and it'd drive a wedge into the existing global standardisation process. That might fit various tech repatriation agendas some governments have.