It is doubtful anyone on these forums would be able to give you a reason WHY Verizon has chosen not to include savings accounts from their list of auto-pay options. It is also doubtful Verizon will provide a reason.
If you have "automatic money that is placed there from my checks", wouldn't it be just as easy to have "automatic money" placed in a checking account? Another option would be to switch money from a savings acct to a checking acct to cover the payment on an "automatic" basis.
Regardless, for autopay I would recommend using a CC if available instead of a checking acct.
The resean is I believe something call Regulation D ( or Reg D) which limits the amount of transfers from a savings account versus a checking account.