The machines in use, from VotingWorks, are precinct tabulators. This means that the voter fills out a paper ballot and the machine scans and retains it, allowing for recount either manually or by alternate machines later on---exactly as you describe. Precinct tabulation was developed to reduce election costs and speed up counting by avoiding the need to transport ballots to a central location and stack and feed them to begin tabulation---they are fed by the voter and tabulated on-site. Almost all precinct tabulators, including those from VotingWorks, are offline and tabulate results to multiple storage media that are physically transported to a totalizing system (typically with paper tape printout as an audit and backup measure, in addition to the availability of the original ballots). This allows election administrators to begin posting a substantial portion of returns within an hour or so of polls closing, rather than the many hours historically required to feed ballots to central tabulators. This is particularly important since polling has found that many voters now distrust results that are delayed in posting and tend to view multiple-day vote tabulation as an indication of fraud, which has added to the motivation of election administrators to have most returns available the night after the election rather than in the following days... a logistical challenge that is difficult to meet without precinct tabulation, given the very limited budget most election administrators have to hire temporary staff.

Unfortunately many people now conflate "voting machine" with direct-recording electronic or DRE machines. While DRE machines became common for a short period after the Helping America Vote Act imposed accessibility requirements that were difficult to meet with paper ballots, most voting machine vendors now offer "ballot marking machines" (sometimes integrated into the precinct tabulator) that allow individuals with special needs to mark a paper ballot using methods like voice feedback. The marked paper ballot can be verified by the voter or another individual before tabulation. Ballot marking machines have mostly eliminated the original motivation for DRE voting and the popularity of DRE voting across the United States has decreased since the shortly-post-HAVA period (2004 election cycle, basically), with many states prohibiting DRE entirely or DRE without voter verified paper audit trail (VVPAT), an arrangement in which all machines essentially function as ballot markers and produce a paper ballot with the voter's selections for the voter to inspect.

DRE with VVPAT can be attractive because it integrates the "ballot on demand" system into the voting machine, simplifying the three-step process typical of precinct tabulation where the voter obtains a ballot from a ballot-on-demand workstation, marks it, and then inserts it into a tabulator. Ballot-on-demand is functionally required by most modern election administrators because it facilitates "voter convenience" models where voters can appear at any precinct (usually within their county), not just at the single precinct in which they reside. The reason for this is simply that, considering the multiple taxing jurisdictions in most parts of the US, a single county election can have hundreds of distinct "ballot styles." Preprinting every ballot style for every precinct is impractical and encourages fraud due to the number of valid ballots "lying around." Ballot-on-demand tends to have a very positive impact on election integrity in this way: the ballot does not physically exist until it is issued by the BOD system which is typically integrated into the pollbook system such that a ballot cannot be issued without marking a voter as voted in the pollbook. DRE with VVPAT is mostly equivalent to precinct tabulation with BOD, but many prefer precinct tabulation with BOD because the majority of voters (those not using assistive technology) mark their ballots manually which decreases the risk of mismarking by a ballot marker.

It's important that American voters understand that non-auditable voting machines are no longer common in the US and are found only in some states, and the portion of votes counted by methods without paper audit trail is decreasing year over year as states replace aging DRE equipment.

Unfortunately, the larger problem with US election administration is not DRE machines but funding. State election administrators usually operate on extremely restrictive budgets. States originally purchased DRE machines mostly because they were the cheapest option that met HAVA requirements, which essentially required many states to wholesale replace their voting equipment on short notice. Most states that use DRE equipment today use it because they cannot afford to replace it. There is very little, even zero, real support for DRE machines other than for the simple reason that states cannot afford to purchase anything else... and this is no longer as true today due to more affordable ballot marking machines, and vanishingly few election administrators are choosing DRE when they have funding available to replace voting systems.

A larger concern in voting security is likely registration and pollbook systems, which are often poorly audited and from small software vendors, once again largely due to the small budgets available to pay for them. In the cases that voting irregularities have been found in the US, they are virtually all a result of defects or limitations of the pollbook system and unrelated to the actual vote tabulation. This encompasses situations like individuals voting multiple times---one voter, one vote is a responsibility of the pollbook system and not the tabulator, which is unaware of the voter's identity as a precaution to protect secrecy of the ballot, constitutionally required for federal and many state and local elections.