It's unclear to me what you're looking for -- a cloud password manager or an offline manager? If you don't like cloud managers, why are you looking at KeeWeb instead of KeePass? KeePass is a solid, respected offline password manager. It's my manager of choice, and can be kept on a flash drive and launched from there, "portable" style. It even supports TOTP codes so it can be used in lieu of Google Authenticator, et al.
However, being offline, it's not as convenient to use as a browser extension like LastPass or Bitwarden. Browser extensions can recognize password fields on a webpage, and optionally auto-fill them from your unlocked vault.
KeeWeb appears to be an enhanced front end to turn a KeePass vault into a cloud vault. But if you wanted a cloud vault, what's wrong with Bitwarden or LastPass? They integrate very well with your browser and are designed from the ground up to work seamlessly with the cloud. And if in fact you
don't want cloud, then just stick with KeePass, not KeeWeb.
Last Password has demonstrated even with good curating, there are always issues. Then there is the added risk when companies change owners or management. Also being proprietary code, you don't always guarantee that data is really not viewable by anyone.
If you mean LastPass, I don't believe the "issues" were ever related to the security of your password vault. The issues were with the stewardship of your master password, stored in LP's cloud, not with the "blob" that is your vault. If you used a good master password, or if you changed it after notice of a breach, you were okay. I believe Steve was impressed with the code behind LP's vault when he got a chance to privately review it, but the security of your master password can be a different matter.
If done right, the fact your vault may be in the cloud shouldn't be inherently riskier than an offline vault kept on a USB stick. After all, you can lose a USB stick. I believe LP and Bitwarden have both done the vault part right, and I have no delusions KeePass' vault code is any better, so I don't consider one better than the others in that regard.
If done right, neither LP nor Bitwarden have access to the contents of your vault. The contents are unlocked on your computer, not in the cloud.
But therein lies the rub -- and your point with regard to changing ownership is well taken. What guarantees do we have that a new owner won't secretly change the browser extension so that when you unlock your vault, the vault's contents aren't leaked by the extension? Or what guarantees do we have that a browser flaw won't cause the extension to spill its secrets?
There's an element of trust there, but it's a question of how much you trust the extension, not an issue with the security of the vault. And with LP's "musical chairs" ownership of late, I'd consider Bitwarden to be the more trustworthy, between those two.
OTOH, that's where the offline nature of KeePass might be more secure -- with no online component and nothing in the cloud and no integration with your browser. But that also makes it less convenient to use.
So take your pick. I trust KeePass for myself, but recommend Bitwarden for friends and family. KeePass is just too geeky for them, and a password manager -- no matter how secure -- is of little use if they avoid using it.