I died on the inside after reading this 😫
If you don't want to solder, you can buy "quick connect/disconnect" terminals and crimp them on. Then you can just slip them over those metal tabs on the new speaker.
You can probably find them in small quantities at your local ACE or Orchard Supply hardware stores. Otherwise, you can buy packs of 100 from Amazon. You can also buy little assortment kits in various sizes.