The reality is that HR 620 is specifically designed to weaken the ADA (see e.g. https://www.aclu.org/other/hr-620-myths-and-truths-about-ada-education-and-reform-act and others), which is why Senator Duckworth is rallying the opposition to this effort to gut the law.
In this particular case, if the changes under the ADA Castration and Business Discrimination Enabling Act (as HR 620 should be called) were enacted, the process would be:
(1) Complainant sends a written notice to Violator "clearly identifying" the problem. Included in this has to be detailed circumstances describing the barrier (to public a accommodation), including the physical address of the property(!) and whether or not a request was made for assistance and whether the issue was permanent or temporary.
(2) Violator has 60 days to send a response describing what they propose to do about it.
(3) Violator now has 60 days to implement their description of whatever they proposed, unless there are "circumstances beyond their control" which allows them to just show progress.
(4) Complainant may now (120 days later) try again. In theory, if "the barrier" is still present and "the same", Complainant can now file suit against Violator. However, if the Violator just rearranged the deckchairs (as it were) the specific details of the barrier would be different, so it's back to step 1.
(5) Having filed suit, the legal wheels start turning, slowly. Typically, at this stage, whatever the barrier happens to be gets addressed and the suit is dismissed as moot before anyone actually gets into a courtroom.... but this time the proposed solution has to be agreed by both sides. Note that no-one really wants to get into court, as there's no money available in most cases.... so the only costs are those needed to pay your own lawyers.
Under the old system, you start at step 5.
The "problem" is that there have been (a few) law firms that have gone around looking for ADA violations and suing the violators, offering to settle for $$$s and addressing the issue, usually with the aid of a possibly-related compliance monitoring company. These scummy law firms use the fact that (for them) the cost is negligible to prosecute the suit, but for the defendant/violator it's expensive, so they can refuse to settle until offered sufficiently juicy "go away" money. Note the ADA has nothing to do with this: plenty of scumbags operate by suing for "go away" cash under many different laws and legal principles (e.g. many slip-and-fall cases, for example).