There is actually no technical reason why phones can't be forced to connect to just one tower (more specifically, one cell).
In fact, cell locking or forcing, as it is usually called, is commonly used by mobile network technicians (like myself) for testing purposes.
It just isn't usually available on end-user phones, most of the time a special baseband firmware is needed to allow it.
There's 2 reasons why it isn't commonly allowed:
1) Mobile carriers AND phone manufacturers don't want people to use it, because it rarely is of benefit to users and causes a big support burden, when people use it and can't use other cells when mobile or when changes in the network happen.
2) Phone manufacturers and baseband vendors charge a considerable sum of money for the unlock license (usually a 4-figure sum per device)
As for using cell locking/forcing to avoid triangulation: this does not work. Even if a phone is locked on a specific cell (via BCCH, PCI or nCI forcing), the uplink signal will still be received by all base stations in range, which means that location information is just as precise as without forcing.
Not only that, the modem in the phone will constantly monitor neighbor cells and report them to the serving cell (these are called Measurement Reports), so the network knows when to initiate a handover.
The precision of location depends largely on the number of available cell towers, and the used technology.
If beam forming antennas are used, highly accurate location information is available to the network.
Beam forming not only allows signal strengths to be measured, but also measures angle-of-attack, giving a reasonably accurate vector of the signal direction. The precision can be as low as 3 degrees (but more commonly in the order of 6 degrees).
With very good 5G coverage above 3 GHz (and especially in mmWave above 23 GHz), the precision today can be as high as 1 meter.
In the very near future, sub-centimeter precision will be possible, should enough beam forming antennas be in range.
This cannot be avoided without disabling the connection to the network.