But then I wonder: what if I need a higher tonnage?
That depends on what you're doing. Tonnage requirements can stack up pretty high when you're not paying attention. Pushing bearings, bushings, and general stuff like that, you don't need a bunch. If you do that sort of work, you will "occasionally" exceed that, but not much. Hanging a 200 pound gorilla off of the handle in a way that exceeds the stated capacity will probably solve the problem.....
If you want to bend stuff, straighten stuff, run punches, that sort of thing, forces escalate quickly.
If I understand it correctly, hydraulic presses dominate the market because hydraulics in general are quite cheap compared to mechanical drive systems.
I wouldn't say that. I think it's because of the economy of scale in the smaller end stuff. up to 20 or 30 tons maybe, for argument's sake. (Including but not limited to junk, the good brands of light presses are included). Hydraulics are more popular because you get the most force and speed per effort applied. So the parts are rolling out of the factories in large quantities, versus uncommon styles.
But if money wasn't an issue, what prevents me from building a rack (or maybe screw) type press that allows for a much finer control?
Nothing prevents that except maybe your willingness to do some engineering calculations to be sure that you're getting what you're shooting for.
On the other hand, if you want the best precision and feel you're gonna ever get out of a press, you could use mechanical mechanisms for sure, but in practice, if you use a hand powered bottle jack, you will retain that feel. Especially if you make an appropriately sized handle. 99 percent of what anybody does on a hydraulic press doesn't "max" the capacity. A shorter handle amplifies the feel of how things are going. The fluid is non compressible, the feel comes through nicely. And doing this at home.... Assuming that you already have the stock, you can get a bottle jack for less than the cost of tool wear making a screw.
I've noticed that arbor presses come with open throats, compared to H frames. Is that an absolute requirement for the typical kind of jobs you do on them?
Arbor presses have that slot (and hopefully a daisy wheel) which keeps the front open, making it fast and efficient to push arbors into and out of round parts that don't lend themselves to mounting in a chuck when performing the required machining. They get used for a LOT of things besides that, and frankly, using arbors for fixturing is just not as common as it used to be. But it's kind of the definition of an arbor press. That's what they're meant to do.
The H frame presses are the most popular for sure, but there's other takes on that too. Not really in anything practical outside of industry, but the H frames are compact for their rating, take the smallest reasonable amount of material to build for their rating, don't take an undue amount of space for their productivity, are very versatile, and are quite popular, meaning that specialized accessories for pressing tend to be geared towards that design.
The arbor press, when used for general purposes, has a (relatively) small pass through area, BUT, it's pretty wide open. That gives a lot of freedom about what you can get in there. On the other hand, the frame is only so deep, and that's a big limitation. An H frame has no depth restriction on the front or the back, but the sides are kinda always there. That makes it awkward to for example, straighten a long bar.
The bottom line is, what do YOU want to do with this thing, and that'll guide you on to what type is best. And what materials might you have access to? You're not going to buy the steel (forget the hardware and mechanisms) to build a press for what you can buy one for. how much space do you have? An arbor press can bolt on to a work bench, but then there it is. Or a big stand. You're gonna be yarning on that lever, and you're going to find quickly that they work best if you're not strong enough to have the whole thing move when you do that. An H frame press (in a hobby use, not in industry) can pretty safely be put on casters to roll out of the way, but it's gonna eat up floor space "somewhere" at all times anyhow.
Your use case, and nobody else's is going to guide the best decisions in this case. Personally, I find the H frame type a LOT more versatile. In the lower end, you're gonna want some "proper" apron plates, some shims/spacers, and all kind of little adapters to get "everything", but it's not bad. Most can be improvised if they're not used enough to warrant making proper tooling.
Within the H frame pressses, I will say that I REALLY prefer the type with a hydraulic cylinder mounted at the top, The bottle jack style has the extra traveling part below the jack, and that, to me, is annoyingly in the way more often than I like. The mounted cylinder type also will "typically" have a much longer stroke, so less apron resets in between starting a push, and finishing a push. But again, what are YOU doing, what do YOU plan to do in the future, what space do YOU have....
