I know I shouldn't take the bait, but here I am with the cheese in my grubby little paws anyway.
If you truly think that people who believe those two things about Doffy are wrong, then the truth is that you have both a flawed moral framework, and bad media literacy.
Let's address them one at a time.
One, Doflamingo was barely ten years old when he met Trebol and the other executives.
Two, Trebol was 18 at the time, Diamante was 15, Vergo was 10 and Pica was 9.
Three, textually, canonically, Trebol says on the page that he groomed Doffy to be a king. Groomed in the traditional, non-sexual sense where one 'grooms' a young person for a role that they will fill when they're older. There is literally a speech about it, and to back that speech up we actively see a flashback.
There is no arguing against Trebol grooming Doflamingo because it is literally and without exaggeration the text of the story. Trebol says "I raised him to be an evil king" and then we see flashbacks of how he does that. It's just literally the canon of what happens between them.
The only way you can try to argue that Doffy wasn't groomed by Trebol is is you truly believe that a 10 year old child is both completely morally responsible for their own actions, and also so manipulative that he is completely in control of all of the people around him even as a child.
In other words, it's absurd unless you believe that an actual child can be responsible for adult actions. And if you believe this about children, then you have a flawed moral framework.
B) Doffy genuinely loved his family.
This all depends on what you believe "genuinely loved" means. If you only ever use the term "love" to mean "pure, selfless unconditional love" then sure, I will agree you, Doffy did not love his family in a "pure, selfless and unconditional" way. I would argue that that's a standard of love that most people fail to meet.
However, we see on the page, demonstrated in canon, that Doffy has genuine affection for and attachment to his family, two things that when taken together, can be described as love.
If Doffy doesn't have attachment to and affection for Vergo and Monet, (he does!) then there is no reason for him to secretly leave Dressrosa and flay all the way to Punk Hazard to try to rescue or take revenge for them. (He does this! He goes to Punk Hazard immediately after speaking to Vergo, without even telling Trebol!)
If Doffy doesn't have attachment to and affection for Pica, then there's no reason for him to threaten people over laughing at Pica's voice. You may not like it, but it is an expression of care by Doflamingo that he threatens people on his friend's behalf.
The care and trust between the executives is so important that Law tries (successfully) to use it against Trebol in the big fight. When he says "Doffy doesn't care about you, he's just manipulating you", whether Law actually believes it or not is irrelevant— what he's doing is using the horrifying idea to manipulate and enrage both Trebol and Doflamingo.
Yes, Doflamingo's love comes with strings attached, but unless you are withholding the term "love" for all but the most pure and selfless of emotions, then it's simply impossible to say, simply from what we see in canon, that Doffy didn't love the executives. He did, we see him act on those feelings in the text.