Does Spectrum Mobile support Visual Voicemail on the iPhone?

Yes. See the following from their support site: https://mobile.spectrum.com/support/article/360001909128/visual-voicemail-for-iphone

My dad has Spectrum Mobile and he can't activate Visual Voicemail. His voicemail tab on the phone app just has a "Call Voicemail" button in the middle that takes you to a voice menu to hear your messages. No list of VMs, transcription previews, and no "Greeting" or "Set Up" link.

It turns out that when he switched from Verizon to Spectrum and kept the same phone, something needed to be reset by Spectrum for visual voicemail to work. It's fine now.