No it does not come with a heatspreader
Yes you should, but do not cheap out on the heatsink, buy something good, like this:

https://my.shp.ee/ohg2Z2g

Corsair & WD Black has PS5 heatsink versions of their Gen4 drives. The prices arent too far apart from KC3000 + compatible heatsink that you might have to swap around if its too high. A lot of these HS says its compatible but didnt factor in if SSD is double or single sided. Corsair & WD gives hassle free compatibility from the box.

Not all heatsink variant is compatible with PS5, so please dont reco incompatible ones

To OP, for PS5 you specifically will be looking at "CORSAIR MP600 PRO LPX" or "MP600 ELITE for PS5" or MSI "SPATIUM M480 PRO PCIe 4.0 NVMe M.2 PLAY" as these HS are vetted to be fully compatible. Fury Reg HS is also vetted to fit so choose whichever you like.

EDIT
Coincidently this perfectly drops in just today https://www.techspot.com/bestof/ps5-xbox-storage-24-25/

IINM Pro HS variant too high to fit PS5. LPX HS version does. Elite use new Phison controller, and sold as compatible with PS5.

S70 blade dun have HS, i know cuz I using one now. S70 (non-blade) is with HS but that doesnt fit PS5, too high. If want Blade need to get 3rd party HS which may or may not fit cuz 2TB above version is double sided.

WD SN850X HS variant is normal SSD that can fit both PC & PS5.

Fury Reg is a good choice, practically its the same hardware as that Corsair.

WD SSD is marketed as an integrated solution so your getting more consistent quality, availability, the price gap also somewhat justifies with having overall better performance than all Phison based Gen4 SSD, and with free cloning software tho this isnt beneficial to you if used for PS5, and generally WD has the better branding name for storage matters and should have faster RMA turnaround (their SSD are Made in Msia unlike nearly all others).

For those who can that bit more for peace of mind, its WD or Samsung.