Paragraph (h)(1)(A), which you did not reproduce,
The Secretary shall prescribe such regulations as may be necessary for the development and implementation of an electronic fund transfer system which is required to be used for the collection of depository taxes.
Browse through the rules prescribed by the secretary of the Treasury and you will find this,
26 CFR 31.6302-1(h)
Time and manner of deposit—deposits required to be made by electronic funds transfer—(1) In general. Section 6302(h) requires the Secretary to prescribe such regulations as may be necessary for the development and implementation of an electronic funds transfer system to be used for the collection of the depository taxes as described in paragraph (h)(3) of this section. Section 6302(h)(2) provides a phase-in schedule that sets forth escalating minimum percentages of those depository taxes to be deposited by electronic funds transfer. This paragraph (h) prescribes the rules necessary for implementing an electronic funds transfer system for collection of depository taxes and for effecting an orderly and expeditious phase-in of that system.
(3) Taxes required to be deposited by electronic funds transfer. The requirement to deposit by electronic funds transfer under paragraph (h)(2) of this section applies to all the taxes required to be deposited under §§1.6302-1, 1.6302-2, and 1.6302-3 of this chapter; §§31.6302-1, 31.6302-2, 31.6302-3, 31.6302-4, and 31.6302(c)-3; and §40.6302(c)-1 of this chapter.
Section 2, which I did not quote here, provides rules for the gradual phase-in of required electronic tax deposits from the 90s.
ETA: To directly answer your question, it turns out businesses are not required to use the eftps website. But we are required to pay by “electronic funds transfer”, and the government decides the methods of electronic funds transfer which are allowed. Section (h)(4) of the above cited CFR links to this document which gives details on the electronic federal tax payment system (EFTPS) businesses are required to use:
(An automated phone system is available in addition to the website)