From a price perspective you can certainly argue this is a fair comparison to make since the list price is similar, but in terms of why the SE exists it's just wrong to consider the SE as the "budget phone" when Apple sells the modern-style iPhone 11 for about $70 more.
The iPhone 11 is Apple's budget phone. $500 is fairly expensive for a budget phone, but that's the position it holds in Apple's lineup. Apple sells high end phones, so yes, even the cheap one is fairly expensive.
The SE is Apple's change-averse-user phone. It also happens to be cheap, but that's a side effect, not its reason for existing. Treating it like the budget iPhone intended to compete against more modern-style budget phones misses the point. It's like calling a sedan a budget SUV with an undersized cargo area just because the price is lower and it can basically be used for the same things, or saying a manual transmission on a modern car is a budget option with outdated features just because it happens to cost a bit less than an automatic.
There's a very substantial difference between the 11 and the SE; the SE doesn't have a small screen and home button because it's cheap, even if they do make it cheap. It has those features because it's specifically targeted at users who like their old-school phone and just want to replace it when it breaks. It's very clearly designed for users who say "I just want another one of what I already have, maybe with newer guts."
The SE is the manual transmission sedan of iPhones. If you want a modern phone from Apple without top-of-the-line prices--the no-frills automatic transmission crossover of iPhones, that is--the 11 is in the lineup for you.
I am not the target market for either of these devices, but I really appreciate that Apple has kept an "old-school" iPhone in the lineup, with fully modern guts, for change-averse users, particularly older ones to whom specs are meaningless but familiarity is critical. I know more than one person in that exact category, and I'm glad there is a phone they can buy when they inevitably break their 6, or 7, or 8, or older SE and need to replace it.