I've done foam surround repairs for various of my audio customers over the years, and no, you do not need to replace dust cap for most surgeries, unless you are re-coning (which the spider has to be re-aligned as part of the process, needs paper shims).
in my case: I use an AF signal gen + one of my oldie transistor amp, set the signal gen to around 30~40Hz while connected to the speaker, adjust the volume until the woofer cone excursion about 8~10mm total (meaning 4~5mm up, 4~5mm down) while the glue is still wet.
Once the VC aligns properly (after doing the said step), turn down and then off the amplifier and let the glue dries.
That's it!
Other than some rather pesky small diameter (40mm dia woofer), so far I've done countless woofers and foam-based F/R speakers starting from around 6" dia all the way up to 15", with 8~12" being the most commonly done so far.
JBL, older Infinity, Mirage, Boxe,etc.
Q.