To answer in the style of the question... TH SXTN WRDS/PHRSS N TH CNNCT WLL CLD B CTGRSD LK THS:
Group 1: HAM
HamBuRGeR (food)
Ham-FiSTeD (clumsy)
Ham aCToR (overacting performer)
Ham RaDio (amateur broadcasting)
Group 2: HEY
Hey JuDe (Beatles song)
Hey PReSTo (magician's cry)
HeyaWaKe (puzzle type)
HeyDaY (one's prime)
Group 3: RE
JuS iN re (legal term, a right in property ownership)
RePoPuLaTe (fill again with people)
ReFiNaNCe (take out another loan)
Re:ZeRo (Japanese short novel series) (solved by @JS1)
Group 4: The suffix -ITE
SoCiaLite (posh party-goer)
MeTeoRite (space rock)
MaGNeTite (iron ore)
ReaGaNite (supporter of Ronald Reagan)
(Okay, I'll write normally!) As for the 'thematic' final connection:
This is not your typical connecting group using the link-words as they are! Instead, in keeping with the puzzle's theme, let's now remove the vowels from the link-words! This gives us the following:
HAM --> HM,
HEY --> HY,
RE --> R, and
ITE --> T
Re-order these and put them together to find that the 'thematic one-word answer' is:
RHYTHM - one of the longest words in English that can be made without any of the 5 vowels A, E, I, O or U!