What are my rights when I have a Sparpreis ticket but can't board an overcrowded train?

The accepted answer that you linked to is concerned with a different case, namely that the train is marked as "please take a different train" in the on-line schedule. In this case, the "Zugbindung" (ticket is valid only for a specific train) is not automatically cancelled and you have to take the train if you don't want your ticket to become void.

If a train is actually to full to leave the train station, the situation is different - the train conductor and/or the federal police (if asked to assist by the train conductors) will ask passengers to leave the train. In this case, you are entitled to continue your journey on a different train. If the overall delay with the next available train is at least 60 minutes, you will also get a part of your ticket price back later (if you apply for this).

In this case, you options are:

1. Just hop on the next available train, telling the ticket conductor that you had to take the next one as due to overcrowding, passengers without a reservation were thrown out of the previous train. They are likely to be unable to verify this, so the better option is option 2:
2. Go to the information booth and get a stamp that the train was overcrowded. This cancels the "Zugbindung" and you can take the next train.

There have been reports of cases in which the overcrowdedness of the train was not properly recorded in the database of DB, which led them to refuse your right to take a different train. The correct and tedious way of dealing with this is to contact "DB Kundendialog" after the journey (and most likely having to buy a new ticket to continue your journey).

In general, this is driven by the practicalities of rail travel.

First let's exclude cases where the passenger has missed the train.

When the carrier imposes a restriction that the transport will only work on a certain run (like nearly all airline tickets), and the carrier refuses to provide transport on that run... then the carrier is obliged to provide reasonable alternative transporation.

Airlines are obviously good at this, because they are also formal. They formally reticket you (because load management is critical, no standees in air). And you need a document so you can exit and re-enter security.

With railway, well, they are also pretty good at this, but they tend to do it much more informally.

At minimum, they may simply tell you to use that ticket on the next available train. What happens then? Most likely the conductor has already dealt with 10 other parties with the exact same story, so that proves it's true. Or if you're the first, it's possible you are lying and were late due to your own fault, but the conductor knows company policy is to let you ride the next train with available (about-to-be-wasted) space, and here you are. It's a distinction without a difference, and certainly not worth making a huge scene.

Also, this is not his first rodeo: he knows the 1:00 train often overbooks so your story is plausible to him. He may even be aware of an equipment problem (car removed from the train; car present but with inoperative A/C) that caused the overbook.

* For Amtrak/Via long-haul, this tends to be a huge production because of the need to book hotel if taking next train, or airline tickets if your travel is time-sensitive or if your multi-day train can be overtaken. Fortunately this mainly happens at hubs: NYP, WAS, CHI, LAX, Toronto which have enough staff to handle a mob.