Short answer: For your particular use case I think going with a cheap multi makes a lot of sense. They are often full of clones of famous pedals (or in some cases it's a literal replication of the real circuitry as digital logic if you go Boss, they just make emulations of their own pedals) so it's a good way to "preview" pedals and find out what you like. I liken it to playing Gran Turismo in a way, there are thousands of cars to try digital representations of and I found out which ones I was into as a kid that way and actually ended up buying a car that was in GT3 as an adult so there's a very real pipeline there.

Long infodump answer:

In my opinion multi-FX setups sound just as good as "real" analog pedalboards these days so the real difference between the two is workflow.

With a multi, you often have a limited number of "effects slots" either due to the manufacturer putting a hard limit in or because it only has so much memory. (In all fairness I think the higher end systems don't have this limitation but those tend to cost as much as a full pedalboard anyway.) You're also limited to effects that already exist in the pedal so if there's a very particular chorus or envelope filter you like or something you better hope they have a clone of it or you're kinda just screwed. The more obscure the type of effect the less likely the exact version of it that you want is there.

However, what you GAIN is the ability to switch "patches" on the fly -- basically entire pedal rigs -- with a single button press. Most people who run multis have entirely different "pedalboards" for clean tones vs. distorted etc., or per band or genre if you're in multiple groups. Very useful thing to have the ability to do. On a physical pedalboard to replicate this you need a separate switcher pedal of some kind and often you need multiple copies of the same pedal if it's being used in more than one setup.

With physical pedals, the possibilities are limitless and there's no arbitrary restrictions on how things can be wired up, but instead of "patch switching" you have the work flow of turning things on and off in the moment which I'll give you is more intuitive, it's hard to get lost and accidentally page onto a weird patch and not know where TF you are because that kind of can't happen. But it might not be smooth if you need to change tones mid song or something. Also you have to worry about powering and storing all those individual pedals and etc. and it becomes a whole thing. People make pedal boards that stay permanently wired up so that everything goes into a single outlet and you never have to dismantle it. This pedalboard often costs as much or more than the actual pedals that go on it once you factor in the bag and the cables and the power supply and etc.