I think Alchemist should be able to make Medicine ckecks with intelligence.

It feels like a thematic thing they should have, and sure your character doesn't have to be a doctor for the alchemy they perform but knowing how you potions/elixirs work on biology would be necessary for making them or at least applying them effectively.

I just want to see everyone's thoughts on this.

I would love that yep.  Just one of a few changes I'd rock for Alchemists.   I've taken expertise with it excplicitly to fulfil the character concept. 

(I also wish they had Primal Savegry on their spell list)

I think it should be Intelligence based for everyone in the first place. Doctors spend years of learning about the body and how to treat injuries/health issues. 

It depends on how the skill is being used.  INT is a pretty common alternative stat to apply.  Class or subclass is irrelevant.

Completely understandable to use the PC's knowledge (Int) to make a deduction rather than intuit (Wis) what is wrong.
From P239 of the DMG:

SKILLS

As described in the Player's Handbook, a skill proficiency represents a character's focus on one aspect of an ability. Among all the things a character's Dexterity score describes, the character might be particularly skilled at sneaking around, reflected in proficiency in the Stealth skill. When that skill is used for an ability check, it is usually used with Dexterity.  Under certain circumstances, you can decide a character's proficiency in a skill can be applied to a different ability check. For example, you might decide that a character forced to swim from an island to the mainland must succeed on a Constitution check (as opposed to a Strength check) because of the distance involved. The character is proficient in the Athletics skill, which covers swimming, so you allow the character's proficiency bonus to apply to this ability check. In effect, you're asking for a Constitution (Athletics) check, instead of a Strength (Athletics) check. Often, players ask whether they can apply a skill proficiency to an ability check. If a player can provide a good justification for why a character's training and aptitude in a skill should apply to the check, go ahead and allow it, rewarding the player's creative thinking.

Ultimately it is up to the DM to allow this, but it is completely accurate to allow.