What does an unregulated linear supply look like/what do I look for?
https://www.eleccircuit.com/unregulator-power-supply/
This is quite a good guide.
High-current unregulated supplies aren't sold commercially much as the switch-mode supplies that you linked to are so much cheaper and better for just about every application unless a large inrush is needed. However unregulated supplies are really easy to make, as Schmitt Trigger suggested.
I would, however, suggest a much larger capacitor than 10,000 μF.
Transformer:-
https://cpc.farnell.com/multicomp/vtx-146-120-112/transformer-120va-2x-12v/dp/TF01609
Rectifier:-
https://cpc.farnell.com/multicomp-pro/kbpc3506/diode-bridge-rect-1-ph-600v-module/dp/SC15622
Capacitor:-
https://cpc.farnell.com/multicomp-pro/mclpr25v479m35x56/cap-47000-f-25v-alu-elec-snap/dp/CA08669
Notes:- the transformer is a little bit under-rated, so it will get hot but that will take some time. The next size up is a lot more money. You could use two 120 VA transformers for a lot less money than a single 225 VA transformer.
The rectifier doesn't need to be 600 V but it won't matter.
The capacitor should give about 2 V of ripple at full load. The capacitor is rated at about 10A of ripple so you should not use a smaller one that would have a lower ripple rating. A 10,000 μF one may overheat as it would only be rated at about 3 A of ripple.
On the transformer, wire blue and brown to live and neutral of the mains supply. That is the only mains-voltage part that has a risk of electric shock, so make sure that the connections are insulated. You can turn on when that is done and check that you are getting around 13 V AC on the other windings of the transformer.
Wire orange to black on the transformer, and connect those two to either of the connections labelled "~" on the bridge rectifier. Wire yellow to red on the transformer and connect those two to the other connection labelled "~".
Wire the "-" connection on the rectifier to the -ve of the capacitor. Wire the "+" connection on the rectifier to the +ve connection of the capacitor.
And that's all. The load (the compressor) can be connected to the two capacitor terminals.
Put it all in a box if you want to. The case of the rectifier can be connected to ground, but the case of the capacitor should not be connected to ground. Do not put any conductive items through the middle of the transformer that connecting to anything both sides.
Tell us if you get it all working.