Just throwing my two cents into the ring since I've read the playtest and built a low-level summoner to include as an npc in my game. I don't have any real play experience, so this is all just theory based on what we have in the playtest document.
I think the idea of sharing actions and health with the eidolon is a cool idea to help tone down the action economy of playing two creatures. I suspect in play that tandem actions will feel a little rigid with just the current options and I'd like to see some feat that allows the summoner to cast a 2 or 3 action spell while the eidolon gets an action or two to work with. Of course we'll probably get more tandem stuff with the official release, just saying I think that's a good place to head with class feats.
I agree, boost eidolon is kinda bad, and I think I know why. On paper the damage boost is more than reasonable for a cantrip that you get automatically, but it costs your tightest resource: actions. You're supposed to be managing two creatures, and the spell is locking damage the eidolon sorely needs behind 1 action a turn. Now this might be controversial based on what you suppose the eidolon's role is, but I think it should fall somehwere around the rogue and monk for damage output and durability, especially since this summoner only gets 4 spells a day and the eidolon is supposed to cover the rest. I really don't think it'll break into their niche without debilitating strike/skills galore or stances/ki.
If that is what it should accomplish, then I really don't think it's there yet. Giving it special access to monster abilities and a wider range of combat options in general will give it its own martial flavor and make it more fun to play in general, since the current eidolon can't do much more than attack and 1-2 limited special abilities from its type. In simple terms, if 1/2-2/3 of the class is martial, that martial 1/2-2/3 should be competent without wasting an action every turn to try and help their sorry dps. And above just basic competency, they should get some of their own wonderful turn-by-turn strategic options that make pf2 martials so much fun.
Evolutions. This is difficult because 1e has such a hideous amount of customization that anything less will feel restrictive to 1e players, but I think the current template system just swings too far into a kind of rigidity that feels very 5e in a game very much unlike 5e. Moreover, making evolutions a class feat thing just feels wrong because every evolution reduces other options like tandem actions and spellcasting options (conduit or otherwise).
I think eidolon building would actually work very well as combat-oriented familiars. The familiar system already has a lot of abilities you might want to put on an eidolon, and all you'd have to do is design a similarly structured Eidolon list with more permanent choices and a growing pool of abilities to buy. Level-lock the choices for balance and that would feel a lot better for those that really want mechanical weight to their build choices. It would also open up class feats for more options to fighting with your eidolon (also a great place for unique, class-defining, extremely fun turn-by-turn martial decisions). You can even keep the base forms like dragon and angel, but just make them "starter packs" of abilities rather than binding multi-level contracts.
Synthesists. I absolutely love the idea of synthesists, and it's kinda unfortunate that they've been reduced to a single 1st level feat. Yeah we'll probably get more synthesist stuff in the final book, but the feat we can see right now is actually mostly useless. You give up all the interesting ways the eidolon and summoner can interact, and you give up spellcasting, and you give up the ability to act as anything but the eidolon (which I'll remind you isn't very strong or very fun at the moment), and the only benefit is that the slightly easier to hit summoner isn't targetable now, which doesn't mean much since they share hit points anyway.
So here's my idea to make the synthesist work: bring back the idea that the melded synthesist is both eidolon and summoner so that the summoner can use their own feats, skills, and class abilities while fused, with the tradeoff that they lose their spellcasting while melded. This is fine and basic for a 1st level feat on a summoner that wants to go full martial in combat, and then you can have class feats later on that make the melding more efficient or open up more options, maybe even bringing back the spellcasting at some point. This would allow a summoner to take a fighter or barbarian archetype and get to go full bio-iron man as it was always supposed to be. Numbers-wise the synthesist would still be in the monk/rogue range and it's still only an archetype so I doubt there's be enough overlap to invalidate any other class. I figure it'd fit a similar niche that the berserker mutagenist alch does.
Sorry for a actual wall of text, half of this is just me trying to get my ideas out of my head. I have big hopes for this class and I really hope the final version captures the wonderful feel of the 1e summoner without being so ridiculously broken. As I test out the class some of these criticisms might fall away, but I'm fairly confident in this assessment.