But not large enough to COVER three pads..
I see your point - let's just say I can't see the point of the point...
1) Cargo does move by (standard) transporter in TOS (and already in ENT). It never moves by shuttle in TOS (or in ENT). So that sorta answers the question already.
2) The largest-ever TOS prop intended to portray an object that would be moved around inside a starship (or moved to or from a starship for that matter) is van Gelder's box, and it moves by transporter. The only things larger are the shuttlecraft themselves, and they are pure tare.
3) If the need did arise to move something bigger, it obviously wouldn't clear any doors - external
or internal. So making the entry method capable of handling greater bulk would change nothing. There's that badass cargo hatch at the bottom of the ship, but things squeezing in through that would get no deeper into the ship. No need for them to, no fault in them not being able to. Stuff coming in through the shuttle hangar doors would also either have to go directly to the holds and stay there forever, or then consist of such small elements that those could move along the interior corridors. With transporters, the latter is easier than ever - no need for a clumsy container or other packaging when the stuff can move directly from the shelf to the ship, item by item!
4) For fine tuning, we could always take a closer look at the transporter room, and notice that the back wall is to be found in different angles in different episodes portraying it. Perhaps it can be moved aside (and the individual panels on the outer, corridor side likewise slid away) to push dear mommy from the pad to the corridor, hand in hand with movie era Scotty even? It's just that this wouldn't do much good, because where could they go?
Also, while we didn't see it happen in TOS, unconscious or hurt people would probably be transported horizontally often enough. (It happens a lot in TNG, but there we unfortunately have site-to-site technology so it's fairly seldom that the regular pads would be involved.) In "A Private Little War", Kirk does yank the bleeding Spock upright, but mainly out of tactical concerns...
And McCoy holds his hand all the time, making this explicitly a load spanning two disks. Plus a tad more, because the Kirk-Spock embrace spills over the one disk while McCoy's little black bag (this time orange) adds to the other.
Timo Saloniemi