The gateway antennas are completely different than the UTs. The gateway antennas, because of the massive amount of data transfer required over a wide range of elevations, they are 'regular' parabolic dishes...I think 1m diameter (but maybe a little bit less?). Those antennas can only track one satellite at a time as it goes from ~west to ~east (not counting the polar sats), and then the antenna has to fast-slew reset to catch its next satellite as it comes up over the ~western horizon. Somewhat related, there are 2 steerable gateway antennas on each sat, which is mostly so the sat can stay in constant contact with the ground over high value regions like the US (as opposed to send traffic to two different gateways)--one dish is locked on the active gateway antenna and the other dish is positioning for the next gateway antenna before ether current link is broken (in other words, 'make before break').
The UTs are an electronically steered flat array, which is really just a bunch of radiating antenna elements on a circuit board (like, hundreds--maybe close to 1000? I don't recall), with a processor that manipulates gain and other things for each individual element such that there's an aggregate "antenna beam" that at the most basic level is more or less what you might expect from a regular parabolic dish. This allows the UT to form a beam that can track a satellite across the sky without requiring physical steering the antenna (like you need to do with a dish) and it ALSO allows the UT to track multiple satellites across the sky, which is useful for capacity balancing as well as system robustness.
The number of gateway antennas required is an interesting thought experiment. For the 4000 satellite constellation There's probably going to be ~200-300 sites around the world, probably uniformly-ish distributed, each with some or many antennas (maybe 2-8?). While the ISL network theoretically allows global traffic to be equally split amongst all of those links, a more practical distribution will have more antennas in higher density/traffic/value regions, with the bounding case being the US, where the infrastructure will be close to imitating total service, and so there will likely be more gateway antennas on the ground than satellites overhead. Conversely very low traffic regions will have a smaller number of gateway antennas (even with potentially a similar density of gateway sites), pretty much just what's required for redundancy, since the user demand won't push the ground or space infrastructure.
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