This isn't a precise 'proof by contradiction' answer (although there are potentially ways to make it so, using descriptive set theory techniques), but more a matter of guiding philosophy.

In the class of second-countable locally compact groups, there is a general dearth of 'universal' objects (e.g. a group in the class such that all groups in the class appear as closed subgroups) and embedding theorems (analogous statements to things like 'every countable group embeds in a 2-generator group'). This is in contrast both to the more special classes of countable groups and second-countable compact groups, and also to the more general class of Polish groups.

By itself, this is inconvenient: we can't hope to prove many general results about locally compact groups by first passing to our favourite universal object and then analysing its structure in detail. But what is being hinted at here is that any given (second-countable) locally compact group is in some sense 'much smaller' than the class as a whole. This gives the possibility to prove some surprisingly strong finiteness properties, once one puts the appropriate caveats around compact groups and discrete groups. This sense that any individual group in the class is small has long been known for connected locally compact groups (modulo a compact normal subgroup, such a group is a finite-dimensional Lie group; especially once you pass to the associated Lie algebra, having finite dimension is obviously a very powerful finiteness property); we are now beginning to understand it also for totally disconnected locally compact groups.