I think it's a question of balance, depending on how strong a position you are.
If you are in a position of weakness, feeling sorry for yourself, recovering from something you are judging yourself on - then you absolutely should start with self-compassion, being kind to yourself.
To continue in this mode would ultimately hit your ambition and drive, so if you are in a bit of a stronger position mentally, a higher gear, then boosting your self-esteem is more important.
What if you find that your ambition is rooted in a lack of self-compassion and you don't need achievement as much as you thought you did? You seem to be starting from the position of thinking you know what you really want.
That would be a very interesting conclusion. It might turn out that actually, you don't Want (in the intrinsic sense) that 80h per week job, or that fancy expensive house. Perhaps a caravan in the woods is enough.
I might be misreading your subtext, but it sounds as if you're implying that ambition is a universal pro.
I guess being self compassionate when you are doing great does not bring a big benefit, but neither does it hurt. However as the article mentions, it's hard to have self esteem when you are not doing that great.
As a PhD student I have this imposter syndrome feeling whenever I hit a wall. Being self compassionate and realising that almost every PhD student hits one or more walls during their PhD has helped me to cope with this.
Loving-kindness while one is doing helps one see through complacency and turning into an asshole. It also naturally start bridging into loving-kindnes for others.
I think one of the points of the article is that you don't need to "boost" self-esteem, which comes on its own. What you are boosting instead are your self-importance and narcissism.
I would draw an analogy to "boosting" stock price. When you are doing everything right, your stock price rises on its own. But you can also do self-defeating things to "boost" it nominally.
I don't know, at the end of the day what are ambition and drive really doing for you? Maybe you don't need it. You can just be you and get what you need.
It feels quite good to live an active, goal oriented life in comparison to a more tentative or even goalless one. I've tried all three and it would appear that the mind and body respond better to the former.
If you are in a position of weakness, feeling sorry for yourself, recovering from something you are judging yourself on - then you absolutely should start with self-compassion, being kind to yourself.
To continue in this mode would ultimately hit your ambition and drive, so if you are in a bit of a stronger position mentally, a higher gear, then boosting your self-esteem is more important.