That's nice "just so" theory, but is contradicted by the reality that the US Secret Service has been known to use concealed Uzi's, and presumably similar compact full auto weapons, in bodyguard roles.
someguydave is correct. Compact automatic weapons make sense for highly trained body guards protecting VIPs when discretion is considered important.
Right. My understanding is that military doctrine is generally to immediately attack when encountering an ambush. Presumably, that will throw your attackers off their pre-planned attack and help you regain the initiative.
So you want a big enough defending team such that you can immediately assault the attackers while also retreating with your VIP simultaneously. For the counter-assault team, you want to suppress the ambushers as quickly as possible (get their heads down), thus the automatic weapons.
As int_19h points out, there are special-purpose weapons made for this (see "personal defense weapons") and they are likely what pros carry.
Compact automatic weapons still usually have either a stock (even the smallest Uzis do), or some other way to stabilize the gun while firing - e.g. the sling is used for this purpose with some MP5K variants.
IMO the most compelling machine pistols are those with small light weight folding stocks, not entirely unlike what what the Uzi had. Machine pistols could only be the optimal weapon if anything bigger wasn't an option, but my main point is that automatic weapons are considered relevant to VIP protection by the trained experts, contrary to the musings of internet commenters.
someguydave is correct. Compact automatic weapons make sense for highly trained body guards protecting VIPs when discretion is considered important.