As I'm sure you're expecting to hear, you get what you pay for. Likewise, there's tons of information online.
In my experience, I'd say you sound like you should stick with rattle cans and work on your technique. Rattle cans aren't the problem - over time you'll learn how to feather with them, how to anticipate when they will spatter, how to work upside down and in between, and you'll learn which products are able to achieve the quality you want for your finish.
If that's not what you want to hear, then the easy answer is: Sure, buy that off-brand gun and give it a whirl. See how it works, maybe it's fine and maybe it lasts long enough for you to feel like you got your money's worth. However, if you're spraying a bike with a finish you want to stand up to the road, you're probably using a thicker paint, and those types of sprayers tend to work poorly with all but the least viscous paints - the product data for the paint itself will determine whether it can be cut to reduce viscosity. If you intend on using paints that are too thick for the sprayer to handle and the manufacturer doesn't tell you how best to thin it, you might try an appropriate additive or cut it with whatever solvent it uses to improve the flow aka lower the viscosity, but then you're diluting the paint and can easily result in failing paint. In general, paint chemistry isn't meant to be diluted or reconstituted.
The other options are far more expensive.
Given the types of projects you describe wanting to work on, given that you are looking to make an investment in dedicated equipment, and given that money is a problem: the first thing beyond the options above that you should be looking for is a low pressure low volume sprayer and prepare to get deep in the weeds.
(If you're thinking of a hobby type gun, I guess that could work, but it's going to take a lifetime to paint anything as big as a bike with the coating thickness you'll likely want since they're meant for stuff on the order of a small damaged section on a window sash or miniature figurines. A bike is not a small object. A model bike is a small object in terms of spray equipment.)
Getting into dedicated spray equipment is a deep rabbit hole. Just keep in mind two things:
This site's rules prohibit product selection advice.
and, as mentioned, it doesn't matter the means - whether you go with rattle cans or powered spray equipment - the methods do the work. Technique and paint quality has the greatest impact on your outcome. Granted, like with most things, purpose built equipment will help a lot.
In all cases, you'll need to consider the equipment's cleaning process and required materials. For a rattle can, you just turn the can upside down and spray until all the pain left in the siphon is ejected. Whereas, dedicated equipment cleaning requirements are model specific.
Being on a budget, you're going to be looking for a LVLP gun - you'll need to do your own research on the position of the paint reservoir and the extraction process as they will have a big impact on number of setups and your ability to continuously spray 360 degrees and complicated shapes like a bike. This equipment is still far from being cheap. Your choice of compressor will need to be based on providing adequate volume in order to maintain a reasonable duration at the pressure your spray gun requires. You'll need means to extract moisture from the line. You'll need to know how to work with air hoses and fittings. You'll need to consider the impact of the noise of the compressor if that's an issue. Keep in mind that spraying with compressed air presents a number of challenges.
Dedicated staged airless sprayers: The thing to bear in mind is the more stages, the greater the ability to atomize thicker paints and the longer spray duration. These heat up as they spray as well, but with these it can damage the equipment. The fewer the stages, the lower the cost, but the faster it heats up.
A note on the pot sprayer on a compressor that your dad used to use when he was a carpenter: essentially the same setup as the LVLP recommendation above, but get ready for a mess and it's historically technically been an EPA violation.
Now all this is assuming you are aware of the overspray considerations once you move beyond rattle cans.
Good luck and have fun!
Read these links for an articles that flush out the upshot of the solution above:
'How to Set Up an Air Compressor for Spray Guns & Painting'.
'Paint Spray Gun - How To Pick A Paint Gun'.
'Which HVLP Spray Finishing System is Right for You'.
Read these for context (OSHA regulations, EPA regulations), and bear in mind that although personal small-scale spraying is not subject to federal air quality regulations, the improved transfer efficiency and material savings versus the HVLP makes the operational cost over the lifecycle of the LVLP equipment significantly less than HVLP finishing systems.
Essentially, if you go beyond the cheapo "finishing system" options (rattle can or an all-in-one sprayer) every piece of equipment will need to be evaluated for the impact of it's performance spec on your entire process ... for example, two things to keep in mind:
Remember, high volume means lots of paint. Make sure you prepare for the overspray and the
Remember, the cost-time-quality pyramid (if you don't have much money and you want quality, you better be ready to make up the difference with time investment).