10 minutes can be one encounter if your party is having to do any amount of searching. So you use your fly spell, you bypass one encounter, good for you. Druids can potentially bypass several encounters using their fly.
The magic broom is a bad example because frankly that item is undercosted. Boots of flying has a more limited duration and generally slower speed, and the carpet of flying is very rare.
How is the Druid allowing the entire party to bypass multiple encounters? Do you mean combat encounters, because the Druid can't allow any members of the party to bypass a combat encounter. Unless you are able to do something like fly up to a balcony and tie off a rope... which doesn't take hours of flight to do. That is multiple encounters with a single use.
Additionally, can you name a way that the Druid wildshaped into a bird can get the entire party to bypass multiple encounters that cannot ALSO be done by a Familiar with flight at level 1 from any Arcane caster? Like, you make the claim "multiple encounters" but I don't know what this looks like to you, what are you seeing here?
And gain a stronger offense, speed, xyz secondary ability (like trip), bonus action unarmed strike.
You mention the bonus action strike, which means you are talking new Druid. Which, first of all, the bonus action unarmed strike is indistinguishable from the trip attack, they are the same thing, and the unarmed strike is NOT a damage ability, it is purely control. Unless You are speaking about a specific homebrew solution and I missed that.
Going back to the playtest druid though you DO NOT get a stronger offense, at least not until level 5. Levels 1 through 4 are identical or weaker offense. Also, you have to consider that many of the wildshape templates offer really weak offense options. For example, the Flight Form gives multi-attack and flyby, which sounds very strong... except it is 2d4+wis mod. Assuming that by level 9 you have a 20 Wisdom that is 2d4+10 or 15 damage. This takes two actions for non-moon druids, which is two rounds. Do you doubt that a full caster with access to 5th level spells can't do at least 15 damage over two turns?
Sure, the Land form with Elemental strikes at level 10 is doing significantly better, they have a solid damage output of 2d8+2d6+10 or 26 damage, and can do that round one with a bonus action set-up, but that is also the exact damage of a rogue with a shortbow (6d6+5 = 26) who can use their bonus action to hide and was attacking at range and potentially even with advantage, while the Druid was in melee.
Heck, a druid who loses their mind can max dex and grab a shortbow to utilize Swift Quiver and do 3d6+15 = 25.5 or almost identical damage to the Land Form while staying much safer.
I'm not saying that 26 is bad damage, but it isn't good enough to make the druid desire losing AC and charging into melee. And that is solely the Moon druid, every other druid is looking at 2d8+10 or 19 damage with a full action set-up. Just casting a 4th level flame blade is better (only a bonus action to set up, deals 5d6 = 17.5, doesn't reduce AC) and I don't think Flame Blade has ever really been considered a stand out combat option for Druids.