There are at least three very different approaches to reconstructing medieval combat that yield different conclusions and assumptions.
1. Tournament-Style HEMA sparring. The gear somewhat limits your mobility (older/heavier jackets and gloves especially), the common heavier gloves are more likely to receive hits that bare fingers would have been protected from by a properly aligned crossguard, the Feder is a mediocre sword simulator (usually improper mass distribution, excessive length-per-weight, often too flexible), and most importantly getting hit is basically inconsequential. The priority is hitting the opponent. If you double, you double. Depending on the scoring system, you can even "win" by points from a double hit.
Consequences: Snipes, overreliance on the "vor" rather than control (of "the center" or the opponent's weapon), reckless actions, lots and lots of hits that aren't clean - either because they were shoddily executed or doubled. Heavy gloves also limit your sensitivity in a bind and your ability to use "angled" grips on your sword. That alone restricts "fencing" and leads to more "bashing".
2. Bloßfechten, usually with blunt swords (currently most prominently done by Dimicator, can also be seen by others, with other weapons
https://www.youtube.com/watch?v=G3wfet9yxsE ,
https://www.youtube.com/watch?v=jWBlV4r2wlQ ). Basically no movement restrictions (somewhat limited by the mask of worn), choice of Feder or blunt sword (handling quality varies wildly). Yes, you kinda don't want to smash your partner's skull, wrist, ribs, knee or elbow, but that doesn't make you all that slower. The far bigger impact on speed is self-preservation: you need to be safe, because you don't want to GET hit. That's way more important than hitting the other guy. Depending on who you're doing it with and what school you're in you may end up hitting the opponent with reasonable force ("fencing to the touch") or you may stop your hit either before impact or even at the moment both you and your partner realize that you could land it. The latter (fencing to the opening) is obviously what is required for "safe and sane" practice with sharp swords.
Consequences: snipes, especially to the hands, blade binds, grabs, grappling, far fewer double hits. A Zwerch at speed is obviously not allowed to land (if not wearing masks), but the Zwerch of which Meyer is so fond is often NOT a very safe action. Gripping technique in this type of simulation varies a lot and is very situational. You often end up standing in guard with a "hammer" grip, but as soon as the sword is moving and a bind is established a "forward" grip (call it handshake if you will) becomes relevant.
3. Buhurt-Style Full-Contact fighting (SCA, Buhurt, Battle of the Nations, M1). Restricted mobility (helmet, armor, gauntlets), wide range of simulators, getting hit with a swing or thrust at force usually won't be very painful, and depending on scoring system also won't yield any points (and I think it shouldn't - you're wearing armor, after all).
Consequences: lots of flailing and bashing leading to a grappling situation (which is really what has to happen against armor if you aren't equipped or able to thrust in a weak spot). No finesse in gripping swords whatsoever due to immobile-yet-vital hand protection. Also not really any need for finesse, you're having a melee, not a fencing duel.
Neither of those approaches are perfect, all offer some insight, yet conclusions drawn from one situation don't necessarily apply to a different situation. Yes, the handshake grip makes little sense in a situation where you don't engage in blade binds. Yes, the "hammer" grip has its place in swordfighting, either if you're not concerned about range and blade binds or if your gear prevents it in the first place.
(dis)honorable mention: There are some who engage in contact "Bloßfechten" with sharp swords. While those sometimes claim that they're the closest you could get to a real swordfight, they too have restrictions imposed upon themselves: e.g. wearing a helmet with aventail as well as bulky gauntlets (and, funny enough, ignoring hits to the armored regions), as well as not thrusting at all. (for reference [GRAPHIC]
https://www.youtube.com/watch?v=7rzLyx779W0 ). The takeaway there is basically "distance is really really important".
Interesting articles for reference:
https://chivalricfighting.wordpress.com/2014/02/18/why-you-should-train-with-sharp-swords-and-how-to-go-about-it-without-killing-anyone/
https://guywindsor.net/blog/2013/10/how-to-spot-the-bullshit-in-any-martial-arts-drill-and-what-to-do-about-it/
http://www.xkdf.org/blog/2016/10/31/triangulation-in-hema