I'll share my thoughts on this scenario from my perspective as a BJJ instructor (over two decades) with a background in the Bujinkan (about a decade) and significant experience in various other systems, including multiple weapons arts.
- I always teach my students that if a fight goes to the ground, top position is better, especially in a street confrontation. The guard is our backup plan in case staying on top fails, not our first choice. The only possible exception is in certain competition rulesets which are designed to equalize the chances for the top and bottom player. Even for people who intend to compete under those rulesets, I advocate a preference for top game because it carries over better to other contexts. Thus in the situation you present, I would always advocate that my students prioritize stopping the attacker's takedown attempt.
- Your description of the defender's response is unclear. You say "a skillful takedown", but then "lands you on your back already locked into the arm bar." If you are the person performing the takedown you shouldn't be landing on your back. Are you envisioning the defender landing in guard and doing an armbar from there? Or are you thinking about the defender throwing the attacker, holding on to the attacker's arm, and then falling off to the side to perform an armbar? If it's the latter, then I wouldn't personally recommend or teach that approach in a self-defense situation (for a variety of reasons). If it's the former, then it means the defender failed to stop the takedown and is at least attempting an immediate counter attack rather than just laying on the bottom waiting to get beat up.
- In a real fight, you don't use an armbar (especially the type of armbar you seem to be describing) to hold an opponent until he gives up. You just break his arm and then immediately move on. If the situation is not one which justifies breaking his arm then you shouldn't be going to an armbar position. There are plenty of other positions which are much better for controlling someone that you don't want to injure.
- If you have an undeployed knife and someone is in the process of breaking your arm with a properly locked in armbar, it's not actually very easy to access and deploy your weapon. Especially before your arm is already broken. It can be done, but it's not easy. (It's a lot easier if you have the grappling skill to defend the armbar while you fish for your knife.) If an unprovoked street attacker has the will and the skill to draw and use the knife while he is being arm barred, then he probably would have had the knife out and been attacking with it in the first place rather than attempting to tackle his victim with empty hands.
- If you are able to deploy your knife while being arm barred, then the achille's tendons are not a prime target. They aren't in an easily accessible position (assuming your opponent knows how to apply the armbar correctly) and you're likely to end up injuring yourself in the attempt to slice them. Much better to just stab your opponent in the leg. Mid thigh will be the most accessible target.
- In any case, unarmed against a knife-wielding attacker is just a bad matchup overall, no matter what art you practice. I've been training martial arts for 40 years and have practiced a lot of knife defenses from different systems and done a reasonable amount of sparring with training knives. I have my preferred tactics for unarmed vs knife, but I'm not going to pretend that any of them give the kind of odds you would want with your life on the line.
A non-sportive context does change some aspects of application. (Not the overall physical principles, just tactical approach and some technical details.)
IMO, the best approach to understanding those details and tactics for someone who wants "street applicability" is to add in some sparring/live training where you change up the allowable techniques and tactical goals. Grapple with hair pulling allowed. Grapple with strikes. Spar two on one. Spar/grapple with training weapons. Grapple on furniture on in cars. Set asymmetric goals (partner A is trying to hold his opponent down and punch, partner B is trying to get up and get to the exit). The possibilities are endless.
In my experience, the primary driver of success in these drills is the time spend polishing fundamental skills in "sportive" sparring. You do need the mental flexibility to adapt to the new variables on the fly and it helps to do these drills to get a feel for what you have to tweak to account for the changed circumstances. (One pleasant discovery is that many of the adaptations for "street" application can actually feed back to make your "sportive" sparring better. For example, being aware of the possibility of head butts can lead you to put your head in a better position for pure sport grappling.)
Bottom line, "sport" grappling and "street" grappling don't have to be at odds. There are specialized techniques you can do for sport which take advantage of the tournament rules but aren't so good for a real fight. But there are also plenty of foundational techniques and tactics which work really well in both contexts. (Even a lot of the crazy tournament techniques can still teach some really good lessons about body mechanics and movement, so they aren't a total loss for someone approaching the art from a martial perspective.)