What Tools do you NEED as a controls engineer?

It depends what you are doing...

Hands on - it would be a variety of spanners, screwdrivers, small hammer and multimeter.
Doing Software it would expensive desktop / laptop (both) and a variety of expensive software packages.
For the occasional bit of hardware it was easier to discreetly use your own tools.
For commisioning - any and/or all the above plus rug, torch, 'baby wipes' and toilet paper.

I recall going to Thames Water site and having to carry in a desktop to commision a dual redundant Allen Bradley system.
Also looking into a wetwell through a manhole cover on all-fours ... they had commenced 'wet commisioning' without warning, with raw sewage entering whilst a ladder was still in position. In those days ladders were wood not aluminium.

Tool we never had nor any training - a magic wand !
A lot of the development software (SW) for specific brands and models of PLCs either does not co-exist with other versions of its brand or with other brands of software. The SW author(s) assumes their SW has the sole presence on the laptop and it acts like it. That almost mandates that each such software package be installed with a virtual machine operating system. The cost of obtaining and maintaining licensed development software is a serious issue with SI's.

I carried a small resin table with folding legs to have a place to set the laptop. Plant guys would laugh at me and ask if I'd iron their clothes, too, because the legs folded up like your grandma's ironing board. I'd always notice the next time I went in, there were couple folding leg tables just like mine here, there and wherever.

Tethering one's phone to a laptop used to be more costly than it is nowadays, but I pay the additional monthly service because I always need to download something from the web and using some else's 'guest network' outside of their conference room is a stumbling block.

I label all devices that have an IP address. I carry a Brother label machine with me. And I label all sorts of other stuff, frequently putting the labels on the inside of the panel, when I suspect I'll be back and will need the same information next time.

I carry spare AA and AAA batteries for the signal calibrators, label machine, voltmeter, flashlight.

Spare fuse(s) for the DVM's Ohms scale.

I've used a battery powere LED lamp with a magnetic base to light up electrical enclosures that don't have an internal lamp.

I have several bungi cords that I use to hold the panel door open when it wants to close up on me when I'm working.

I carry a couple styles of USB cables and little kit of USB adapters - some instruments are now using USB as access ports for their config software.

I use one of those 3 pronged 120Vac LED outlet checkers for wall outlets. I've found numerous ones with no earth ground or L1 and L2 reversed. I could use a DVM to do the same thing but checker is small and convenient. I blew one up one time when the outlet was wired with 240AC. Better a $5 checker than some piece of equipment more costly than that.

Masking tape to mark cables and wires that have to come off. Not permanent but cheap and easy to write on.

extendable magnet to pick up those machine screw nuts that fall wherever.

Extendable mirror (on a swivel joint) to read the model number on the transmitter that installed up tight against the cooling water pipe.

Hollow ground screwdrivers that fit into bore holes of the terminal blocks and are ground to fit the screw heads.
Interesting - 2 engineers accounts from different eras - we could go on:

Junior Hacksaw for re-terminating armoured cables
Spare laptop lead to wire to terminals to power off the control transformer
Not bungi but a small selection of rubber bands - to hold panel doors open or closed, or to wrap round plier handles to hold you work like a small portable vice
Small scissors or sharp blade knife

My own electronic analogue meter, in the battery compartment had a very small reel of fuse wire for those of us who would 'blow it' in the middle of nowhere.
You must conceptually understand how the process you are controlling works.

You must understand how changes in the field report back through sensors to your controllers, and how outputs on your controllers drive something to create a change in the process. Be able to select the best sensor or actuator for the application. Recognize when it's a bad design and recommend better placement or different control type.

Most importantly you must be able to mentally bridge the tremendous mountain of BS to be able to translate sequence of operation to something that will programmatically read inputs to understand where the process is and write outputs to create a change and bring the process in line with the desired outcome. This is the most challenging hurdle IMO, all of your computer troubles, connection troubles, software issues, different programing environments, etc etc must fade into the background.

These are the most important tools you can bring to the table.
Main thing: A powerful laptop with my programming VMs.

Actual tools: Screwdrivers especially small terminal drivers, sidecutters, pliers, strippers, crimpers, knife, electrical tape (red and black), tarpaulin, others only if I can't get an electrician.

Foldup chair and small portable table!

Meters: Multimeter and Mr Signal MR9270S+ signal generator. Would like a cheap oscilloscope and a nice process meter.

The bits and bobs that no book will recommend:
Proprietary PLC cables and Amsamotion knockoffs thereof, every type of USB cable including an extension cable, USB power pack, all manner of monitor cables and a spare mouse/keyboard, 24VDC power supply with convenient breakers and terminals, USB-serial converter (Startech's current one retains COM port number which is nice), serial cable, null modem cable, DB9 to terminal adaptors, patch leads (1 x 50m, 1x20m, 1x5m and several shorter ones, as well as one awful 10m one of last resort to remind me when I need to buy new ones, cheapy unmanaged switch, wireless cell modem, power extension cords, multibox, double adaptors, dymo labeller, vivid (sharpie) marker, ruggedised external SSDs, a range of glass fuses, a few relays and some terminals.

Supreme prize for most utility per dollar: a female - to female patch lead joiner.