I have an old Logitech PlayStation style controller that doesn't have the best compatibility with games. There are various programs that allow you to map keyboard keys to buttons or emulate Xbox compatibility, but I'd rather have most games work with my controller of choice straight away. I'd prefer a non-bluetooth wired controller but does Microsoft even make them anymore or are they only available now from third parties? What would you recommend?

Xbox One controller? At this point, it's probably the best you can get. Most controllers tend to be utter shite.

Xbox 360 wired or wireless both work great. The PS4's DS4 controller is also very good but requires extra steps to get it going as it's not officially supported.

It seems there are many that prefer the official Microsoft Xbox 360 controller to the Xbox One model (something to do with the analog sticks, triggers and bumpers), but unfortunately the cost of acquiring a brand new wired 360 controller is quite high nowadays.

Forget that and get an 8bitdo controller. Four or so controllers in one chipset.

Or if you're desperate, a Logitech F310 would do.

This!

Xbox controllers continue to have some of the worst d-pads I've ever come across.

What's your OS if you dont mind me asking? Xbone Controllers only work wired in Win 7 for example.
I have been using an X360 pad since 2011 and while i'm generally pleased with it, that d-pad is a pain especially for fighting games.

Windows 10.

In that case since your main priority is plug and play, Xbone controller is your best option. Since you don't want wireless just get the package without the receiver dongle for a lower price.
Alternatively i will also recommend the Dualshock 4 but that can be very hit and miss. Only recent games have PS inputs but even that is rare.

I have the Wired 360 Controller and an XBox One S controller.

The One S has Bluetooth if you consider that a plus, but lacks the hard wired cable in exchange for a cheap Micro USB connector. Other than that I would suggest it over the wired 360 controller.

https://www.walmart.com/ip/Microsoft-Xbox-360-Wired-Controller-Xbox-360/14962346
Only 1 left!
$148.48

Why not just get the Xbox One S controller?

Get it from Walmart, test it out, if you think it sucks just return it.

Walmart.com currently has it for $48, so load that on your phone, and ask if they'll price match it, in-case it's more in store.

I second the choice to get an Xbox One S controller.

After my old XB 360 controller finally wore out, I recently got and prefer the XB1 S controller over it. I rather like the improved d-pad, triggers, etc. Darksiders, Hollow Knight, Carmageddon 1 (using the DOSBox keymapper), Way of the Samurai 3, King of Fighters 98 (much improved d-pad for fighting games, though still not quite as good as SNES or PS1 d-pads), and others all played good on the XB1 S controller.

That said, I got really lucky and caught one on sale for $40 instead of the usual $60. I've been using it about a month now and i'm pretty happy with it.

Does the bluetooth signal from the Xbox One controller still broadcast even when it's connected via micro USB cord?

OK, I just tested this.

Connected to wall USB power = Yes, it still broadcasts.

Connected to computer for gaming = Nope. The light stops blinking and stays on as soon as it's connected, even if you turn the computers Bluetooth off.