Elizabethan ships would not pose an insumountable problem, if you make a few modifications, I think. The problems are 2-fold: a) lots of windage from the 1500-1600's era fore and stern castles, and b) low efficiency sails. This will make progress to windward difficult to achieve if the ship is built exactly like the real ones. But, there are things that you can do to surmount the problems that will not greatly detract from the appearance of the ship. If you don't care about sailing to windward, then build your model just like the real ships. I am biased toward windward performance, or perhaps more accurately, biased towards not having to walk to pickup my model, but rather have her sail right back to the starting point
I'm both a sailor and a pilot; I find explaining the forces on a sailboat easiest to understand if I treat them as lift & drag forces, like the forces on an aircraft. This approach is not the traditional one used by shipwrights, though I've seen websites for racing sailors and yacht designers using the aero terms. Hopefully my terminology won't pose an excessive obstacle to understanding.
Wind speed requirement:
Windage (drag of the wind) from exposed hull will make progress to windward difficult. The tall castles on early ships is an impediment, no question. My bottle topsail schooner has a lot of windage since the bottle provides much more buoyancy than is needed to support the rig and RC gear. 3/4 of the hull rides out of the water - adding more ballast to sink the hull to a more realistic freeboard would make the boat very heavy. However, even with a lot of windage, she beats fine in a strong wind; With enough wind, the sails generate enough lift (=force) to make the ship speedy. In that case, the finkeel develops a lot of lift, combating the tendency of the hull to simply blow downwind. So, the first requirement for working to windward is enough wind.
Low wing-loading keel requirement:
The more surface area in the finkeel, the lower the lift/unit area needed to generate enough lift to keep the boat from drifting downwind. With a low enough wing-loading finkeel (=big finkeel), the ship will perform satisfactorily in moderate breezes. Any ship with a lot of windage will benefit from a large finkeel. Thin width (chord) fin keels, used for footys and racing sloops, have a high wing-loading. As a result they only work well if the ship is fast - since these boats are speedy, though, they can use high wing-loading keels. Squareriggers are going to be slower than racing sloops, usually, so they need a low wing-loading keel. My first keel for the bottle topsail schooner had a footy-sized chord. It failed miserably. My 2nd was about 2x footy chord, and it worked better, but still not well enough to make reliable progress to windward. The final finkeel design, which works well, has a chord of about 80% of the length of the hull, and a depth about equal to the hull length; that's a lot of surface area, more than most modelers normally see, I'd guess. A keel this big has a lot of surface drag, so the ship will not be fast in light winds, however. The 2nd requirement is a big enough finkeel.
Flat sail requirement:
Baggy sails don't work well when you are trying to beat. At the low Reynolds numbers our craft operate at, a flat sail (=low camber airfoil) works better. The paintings I've seen of Elizabeathan ships show baggy sails. If you want your ship to work to windward, though, you will probably have to replace the baggy sails with flat sails. My ships have sails made from Tyvek - it's easy to work with, sheds water, and allows flat sails. As a side note about real ships: It's easier to make flat sails out of cotton canvas than out of flax canvas. One of the advantages the yacht America had over it's British competetors in the famous race was the flatness of the America's cotton canvas sails. The 3rd requirement for a Galleon is flat sails.
If you can live with a big finkeel, flat sails, and choosing to sail only on days with enough wind, an Elizabethan ship model should sail to windward fine. She probably won't ever do as well as a lesser windage hull, but I think, based on my high-windage bottle boats that the Galleon should still provide fun sailing (w/o walking).