bp122
Expert at Jibber-Jabber
If you are talking about breaking down a full sheet, as @paulrbarnard says, track saw is the right tool unless you have a £3k panel saw.Bandsaw - handy to have. But how handy when its needed to split 8x4 boards ?, not handy at all.
What you described there is a fully kitted workshop, is not what the OP was asking for anyway.So yes, you need a tablesaaw, as well as a bandsaw, as well as a planer/thicknesser, as well as a cross cutting machine like a powered mitre saw.
Drill stand or pillar drill is nice too, as well as a morticer and very handy to have is an oscillating bobbin sander.
Mitre trimmer is very nice to own.
Router table - Thumbs up.
And if you have a track saw with an MFT setup or parallel guides or both, you can do 95% of what a small to medium table saw does plus what a decent mitre saw does (except maybe for really thick stuff)
For a person just starting out, tablesaw can be a restricting purchase.Huge pile of other bits most of which will see use once a decade, but nice to have all the same.
But No1 start off with a tablesaw.
Not saying your experience is invalidated, but saying it may not be appropriate for a new starter.