Uh oh, you have a Zinsco...and that means that you'll have to replace the whole panel to have a hope of doing anything else to improve the situation
Atop the known woes with the use of "old technology" aluminum wire for general-purpose branch circuits, you have a bigger problem to fix: that Zinsco panel of yours. Given its penchant for starting fires due to breaker-to-busbar connection burnups, plain miscalibration of breakers (leading them to not do their job when called upon to stop an overload event), and even violent blowouts of the breaker casing during high-energy trips, the Zinsco breaker line is not worthy of the title "antique"; instead, that panel needs to be sent on a one-way trip to the smelter. Never mind the fact that Zinscos can sometimes not turn the power off when you move the breaker handle to OFF, rendering any attempt at servicing an electrical system "protected" by a Zinsco panel hazardous at best!
Fortunately, this looks to be a rather trivial job as far as service changeouts go, since your panel was surface-mounted into an unfinished space during the original install. I would get a 200A, 40-space or 42-space, main breaker, NEMA 1 (indoor) panel as a replacement for this, along with appropriately sized breakers for the various circuits in the panel. I would also point this out to the seller, though, as they sold you a house with what basically is a known defect in it, unless somehow everyone involved was not aware of the hazards Zinscos pose.
Note that when replacing the panel, pigtailing the aluminum wiring is not necessary to attach it to the breakers, as breaker lugs are made of plated aluminum to begin with, and have always been safe to attach to aluminum wire provided correct installation techniques are used; however, since you have aluminum wire, it is imperative that the lugs on your new panel and breakers get torqued to the correct specification values for that make and model of loadcenter using an inch-pound torque wrench and/or torque screwdriver. This is also a new Code requirement in the 2017 NEC (see 110.14(D) for details).
Once the panel replacement's out of the way...
You are indeed correct that there aren't very many good options for GFCI protection in an aluminum-wired house, given that CO/ALR rated GFCI devices seem to be scarce these days. Once you get the panel replacement out of the way, the option of GFCI breakers opens up to you, although with a caveat: most locations in a house need AFCI protection as well, and multi-wire branch circuits must be protected by a 2-pole GFCI; however, 2-pole dual-function (arc fault & ground fault) breakers are not made at all, so you'll have to use an AFCI breaker with GFCI receptacles in that situation, or use an AFCI breaker and install a GFCI "spa panel" inline with the circuit.
If you decide GFCI breakers are not your cup of tea, or cannot use them for some reason or another, then you'll probably need to use extensions to deal with the extra space the pigtails take up atop the bulk of the GFCI itself. Using a surface-raceway starter box is probably the easiest and most elegant way to deal with this, although if your boxes are metal, you can use an extension ring instead; said extension rings are also what you need to use for metal boxes in unfinished spaces, where a starter box can't be used.
As to the actual pigtailing repair process, you'll want to use Alumiconns for it (unless you have an electrician qualified to use COPALUM working on your behalf). These are basically mini-me versions of the mechanical setscrew splice connectors often used for larger wires (larger than 6AWG), which means that they work quite well for aluminum to copper and even aluminum to aluminum connections, but take up quite a bit of actual room inside a junction box compared to a wirenut, and require a torque screwdriver to install correctly.