This blog is a bit dated (2015), but it analyzes a single Hackmaster combat session and concludes that fears almost kept them away from a system that made combat dynamic and exciting. It's worth a read, albeit a small sample size that isn't analyzed at higher levels where players generally have more character options.
As to converting Hackmaster initiative to 5E, I actually considered it because it's more complex than D&D, has a more realistic approach to combats, weapon choice matters, and folks have to pay attention the entire combat (less playing on your phone while others are flipping through the PHB to find out what they want to do). That's closer to OSR design and the realism is appealing to those who like more crunch. However, it all relies on a system where everything already syncs (e.g. defense rolls to deflect attacks, rules on weapon reach, engagement, etc.) And that's where I couldn't make it work. I couldn't get a "count up" system working in D&D without tweaking other rules. I felt like I was trying to redo the entire game. One tweak led to another.
Also note the blog author timed the combat at 48 minutes. For a beginner party against 4 kobolds, that's long to me, even if you're learning a new system and having to look up rules. It gave me the shudders thinking back to some 3E high-level battles that took 3+ hours. My
Lost Mines of Phandelver first battle took at best 5 or so minutes, and my current campaign system (Dragon Age/Fantasy Age, taking a 5E break) has low-level battles resolve in 15 minutes or less, with some of that me setting up a grid map battle. This could be player bias and have nothing to do with the system. For example, I envisioned getting a handheld counter (click once, number goes up) and using poker chips (you can act again in 6 segments, put 6 poker chips by your character sheet).
So I don't have a good answer if it can be done. There was too much to do was my excuse. I ended up working up a "weapon speed" initiative system that my groups used for roughly 5 years before I moved back to the simpler set-turn system of D&D.