I think the issue is that while you don't want a team to be forced to play a single style of play, you do need to be able to go full throttle with a style if you're going to have one.
We either need a top two lines that are offensive forces, against any opponent, and can rack up the points consistently (with a bottom six that can take out the opposition's depth) or you need to have enough offensive pieces in your bottom six that they can reliably score.
They played Beagle as a fourth line player because they acted like we had a powerhouse third line that could relegate a 'competent' third liner to be an elite fourth liner.
Sandwich Beagle between Mojo and Bura, Williams or a player of that offensive calibre and you'd have an elite top eight, offensively capable ninth (Beagle), and a potentially great fourth line (Williams, Winnik and probably Stephenson) that might chip in goals from time to time, but knows its role is as a shutdown line.
If you can get Stephenson up to a reasonable, consistent NHL level early on then he'd likely be an upgrade on Richards in terms of overall game, which leaves you just one spot in the regular roster to upgrade. Per Khurshudyan quoting Troy Mann of Hershey, the main issues they've got with him are point production ('pass first' instinct) and consistency: having played a lot on checking lines and the PK, while managing to score and rack up some points, he seems like the obvious Richards replacement 4C to start the season to me. And if he doesn't work out, he'll be a massive upgrade on Galiev and Latta as our depth in the press box.
Worst case scenario is that Beagle, with two great wingers, doesn't lead to as much offense as you'd like (I think we need to think of it as a first line, two second lines, and a third line), so you move him to the fourth line and either trade in a 3C, or move Mojo back to Center and trade in another winger. The most acquisitions you've made is two (plus bringing Stephenson up) all season.
Genuinely, I don't think we need to do more than tweak. We were probably 90% there this season - but those 10 percentage points are what killed us. The issue is who you can bring in that's cheap enough, who is consistent and experienced, and an ES producer. If Chimera had maintained his scoring ability (let's not forget he had 20 goals and 40-ish points) in the playoffs we wouldn't have had an issue. If Williams had been his previous playoff self, all would have been well. Whoever we bring in has to be able to perform consistently in the playoffs - and it's clear that's not something you can predict. I mean, Mojo outperformed most of the team, Oshie was a playoff God, Orlov became a stay at home Dman - these aren't things you can predict, especially when bringing someone into a new system.
One thing that has to be better next year is the coaching, though. Trotz and co seemed very bad at understanding what worked and didn't, and why, and terrible at adjusting to the other team. Nothing else will matter if that isn't addressed.