Keeping in mind that the Uzi was concealed in a briefcase, making overall length a limiting factor, the choice is not strange for the early 1980s.
The Uzi has a ten inch barrel despite its very compact size, and a controllable 600RPM. And there’s no rule in real life that says it has to be fired on full auto.
The natural competition that comes to mind would be an MP5K. However it has no stock, a 4.5 inch barrel, and a higher full auto rate of fire. That variant had also only been introduced a few years before.
The USSS did eventually adopt MP5A3s/MP5A5s and P90s but those are carried more openly with less emphasis on concealment, at least within the restraint of a briefcase carry like in the Reagan era.
Other than the Uzi, many weapons would seemingly be too large for the desired concealment (carbines), too foreign (Skorpion machine pistols for example), or straight up inferior (MAC-10s). Of what’s left, the Uzi is not going to be outlandish in comparison.