IMHO solvents are strongly recommended because there is lingering residue left behind from the old surround material and adhesive. It breaks down into a tacky oily consistency and even though it may not be visible its there. The manufacturers recommend using alcohol too, and they are in the business. Why chance it? My first re-foam job didnt go quite so well after a couple days of regular use the adhesive started letting go and the foams began lifting because I failed to clean the baskets properly by skipping that step in my preparation. Theres nothing worse then doing them and having to redo them!! Ive done a couple dozen pairs by now and all except the one were a success.
IME, you are correct in the comment of the old surround material - it is a polyurethane foam which does break down. But most of the time this is very easy to remove without using any solvents precisely because it is crumbling off. As above, slow repeated passes with fingers, scraping or lightly applied blades will take that all off, leaving the old glue.
ON the cones, the old glue usually doesn't pose a big problem as the glue line is quite thin, and if too difficult to take off, can be left on. This is for the vast majority of refoams. The glue sometimes comes off because it has separated, and if it comes off without much of the cone surface, it can be removed without solvents. new glue and surround usually stick quite well if sufficient but not excessive glue is used, AND proper clamping pressure, AND proper curing temperature and time is allowed for the glue to dry.
Same for the old surround and glue on the frame, but it is usually easier to remove this because of access. Repeated scraping normally is all that is needed to remove the surround remainder, while the glue takes a bit more. Some are very anal about getting to really clean flat frame surface but its is not always necessary. This is the only area where some solvents might/could be used.
You can see below that some cone materials may not be compatible with solvents, and for most of us, there is no "hidden area to try for compatibility".
here are pix of the first cone ruined by alcohol to clean back-side foam/glue residue. note the separation of front face "film" from cone paper and the warping of the cone.
still dithering on front vs rear attachment of new surrounds - any more thoughts? especially regarding resting position of cone when no surround is attached.
Obviously, this cone is / was not compatible with the alcohol - it is a nitrile cone with an apparent laminate / layer coating of a plastic (for damping/stiffening or weather proofing) and both appear to have reacted badly to the solvent exposure.
If it were me, I'd find a nice place to use these as magnetic paperweights (caution leaving any magnetic financial or ID cards nearby) or for experimental purposes, but they won't be usable in this condition for your speakers.
The good cone can still be refoamed properly if the cone is its normal cone shape without any waves. Any waves will misalign the voice coil when glued up and the misshapenned surround will not respond with linear motion when used.
Edit: keep forgetting the OP did buy good used replacement woofers. Have they arrived yet?
IMO, use a nice sharp pointed blade to cut, from the front, the old surround out at the precise outer edge of the cone. This will take a steady hand and a magnifier will help you to see the joint. Then clean off the cone edge back of any residual surround without using any solvent. Clean with a dry swiffer to remove any dust, then run your clean finger over the back cone edge to remove anything else. Clean frame similarly.
If you have the test tone in your source, hook that up to a small amplifier, and hook your amp speaker wires to the driver securely. Gently apply amp volume till you can hear and see the cone moving and there is no scraping. Do not run in this mode for long or very loudly as the cone edges are not supported.
If it were me, I'd dry fit the new surround on the back of the cone and the frame to check for proper fit. If a little tight, the surround sometimes can be slightly stretched in all outward directions gently, but that usually is not needed. Then I'd suggest taking your surround glue with a Q-tip and applying it sparingly all around the cone edge with just the barest of glue on the front part of the edge - this should seal the front/top coating to the cone to prevent it from lifting off as the coating is not extending to the surround now. Let this dry for about 2 hours - don't worry, the glue normally will dry clear or black depending on the glue.
Then I would apply glue to the cone back circumference in a width enough to cover the inner flange of the surround - spread thinly and evenly within 5 minutes. Do the same with the surround inner flange, upper surface - the top side to contact the cone - but with even less glue, really thin but even. Try not to get any glue on the surround roll but don't be concerned if a little is there. Then a few minutes for both glues to get tacky.
This is the hard part - stretch the surround under and around the cone edge (just like you did when you test fit the surround dry), line it all up visually around the cone edge, then turn the driver over and press the surround onto cone firmly. I know access and visibility is limited so you might have experimented with various small thin flat or curved implements to reach in there, like when you were cleaning the cone edge. You might find something like tools used to trim or push cuticles around (in the nail care area of drug store or Walmart/Target) helpful, or curved hemostats, or the like.
Even and firm contact is important to allow both surfaces' glue to bond without air gaps - remember, less is more here, excessive glue will ooze out too much (a little is normal) and should have a still workable and movable ability for at least 10-15 minutes before setting up and not being moveable (just in case alignment is slightly off). Now let this dry for at least 6-12 hours.
Now using the test tone here is what I do - gather some large clothes pins or medium size binder clips, enough to cover at least 4-8 spots around the frame edge. Turn on the tone at low but audible level. Then clamp one outer surround spot to the frame, then the exact opposite. The tone should be clearer (less flapping slap) - then clamp 90 degrees from that (12-6, 9-3 like clock) avoiding the mounting holes. Then go 45 deg. and do the same again. Once fully clamped and the test tone sweetly purring without rubbing noises, turn off the tone. Then remove 2-3 clamps in a quarter region (up to half circle if needed) and then lay down a thin but even layer of glue on the frame, then press the surround evenly onto the glue. Turn on the test tone lightly to ensure still clean output, then turn off, reclamp. Repeat in sections for remaining surround area. When done, your test tone should still be clean sounding and you are completely glued up. Let dry 24 hours.
Any clamp marks will be covered by the outer cosmetic gasket, assuming you saved that or it came with your kit. If there is no outer gasket, let dry for 1-2 hours then carefully remove the clamps. If you are really worried about the cosmetics without a gasket, use a suitable curved piece under the clamps instead to apply pressure. The binder clips will apply a lot of pressure.
Then glue the gasket onto the outer front edge of the surround if you have it. Let dry for at least 12 hours. You could do this at the 6-12 hour mark of the surround drying.
Hopefully, after 24 hours, you will be done, test tone still clean, and now you are ready to hook it back up. Hook the wires back up in the polarity they were (you had notes or a picture showing the wire color to which terminal, right?), and reinstall into cabinet. Be sure there is a good seal from frame to cabinet.
good luck!