Is the secret to landing a summer internship applying as much as possible

Applying to internships in multiple sectors (banking, consultancy, accounting, actuarial etc.) and I feel like I'm hitting the brick wall.
I've only got three rejections so far out of the
20-25 applications l've put in but l'm quite concerned that I won't get anywhere.
I've been told there's often not a lot of science or a methodical approach to these applications (such as with their online tests) and to be honest the more applications I do the more it seems the whole process is random (and the same for each firm).
Is the secret just to apply to as many positions as possible and hope for the best?
Does it help to apply early?
I'm trying my best but it's hard to applications and tests alongside uni work (especially if you've never done applications like this before), l've only managed to muster
20-25 so far but should I aim for 50-100 in a quicker pace (as it's now November, or do more positions come out in the new year?) I do feel you get a hang of the process the more applications you do, but in some ways I regret applying to some of my top choices first with little experience of the process.
How should I view it if a firm takes a long time (weeks) to get back to me after the initial applications or video interview stage?
Any advice helpful as I'm not used to this process.