The new checkout process model is so far only compatible with Portal template development model. For other development models including ASPX + Portal you have to use the Checkout Process web part, even though it is currently marked as Obsolete.
It is possible to customize the checkout steps of this web part and merge them into a single step, but such customization is much harder than with the new Document Wizard Checkout. Some of the controls used in the old checkout may need a lot of customization to allow single-page checkout.
If you decide to go this way you will need to turn on the checkout settings related to this web part using this SQL script:
DECLARE @SiteID INT = YOUR_SITE_ID;
SET KeyIsHidden = 0
WHERE KeyName IN ('CMSStoreAllowAnonymousCustomers', 'CMSStoreUseExtraCompanyAddress', 'CMSStoreRequireCompanyInfo', 'CMSStoreShowOrganizationID', 'CMSStoreShowTaxRegistrationID') AND (SiteID = @SiteID OR SiteID IS NULL)
SET ElementVisibilityCondition = NULL
WHERE ElementName = 'Configuration.Settings.CheckoutProcess' OR ElementName = 'Tools.Ecommerce.ChackoutSettings'
Afterwards the checkout settings will appear in Store configuration.