I've seen others comment on how the shield was used as an effective secondary weapon (and it was and would still be). However, since the question is "how can shield only combat be a good idea?" - I will add some real-life historical examples of how people tried (and in one case kinda succeeded) in using the "shield" as both weapon and defense:
These we know were real and actually used in at least one duel/street fight using the standard sword & buckler style. The fighter in the linked articles describes the shield & their usage:
[Shields] whose light was cast only forwards by their being made with an iron plate on that side towards the holders, so that their bodies remained in darkness, had not only the advantage of seeing him when he could not see them, but also dazzled and offended his eyes with the many near lights, which made him mistake those objects that dimly he discerned.
However, there are some weird combinations of these that not only had a place for a lantern but were covered in blades:
The above is a real piece from the 16th century, currently at the Kunsthistorisches Museum, Vienna
Yeah, why use a sword and shield when you can have both in one convenient package:
yes the blade is extendable. yes, this means that the weapon in assasin's creed actually has some basis in reality
Okay, there was a point in history where it seemed like they were trying to put a gun in everything. The odd part is though this one "failed" as a hand-held weapon, sources vary but most describe it as too hard/cumbersome to hold & aim by hand. It did have success when mounted as anti-personnel weapons on ships and was a precursor to some modern riot shields and our modern mounted gun shields:
Italy, circa 1540 - Higgins Armory Museum
To add to the idea of gun shields, though modern versions are more a shield with a gun mounted behind it. At least one science fiction show, about a mad man in a blue box, has thought of how it could be adapted in the future to actually be more like its 15-16th century ancestor:
It's all about the combinations
This is certainly stretching the idea of "shield only combat" but all of these are first and foremost shields. So if we allow an extending of the idea of shield only combat to include combination weapons (and use a little hand-waving to bypass the flaws) - its certainly possible.