Rather than tempering your enthusiasm, I'd add to your message (after all the enthusiasm and thanks/appreciation is shared) that you plan to compare offers and respond with a decision once you've gotten responses from your other applications. If you know a date that you'll have those other responses by you can include that date, if not it's fine to be generic (and anyone should understand that a student applying for grad school does not control the timeline of other schools). You could ask if there is a deadline to respond if this isn't already clear.
It's possible this will sadden the professor a bit if they're very excited to have you, but frankly, they have no ownership over you no matter how much they've "stuck their neck out". You owe them an appreciation for their efforts, not to deny all other offers. Most likely they will understand this; if they are the kind of person who is deeply offended by this, though, I would urge you that you do not want this kind of person as a supervisor if you can avoid it. Therefore, you do not need to worry about their reaction too much; if they have a normal person reaction, they will completely understand that any applicant should consider all their options before making a decision. If they react badly, well, you now have dodged a situation that would only get harder and harder to get out of if you found out later in your studies what kind of person this professor is.
A final caution would be that at least where I am from in the US, official offers need to come from the institution, not the individual professor. An individual professor may very well have strong sway, but you need to have an official institutional offer in hand before you accept it and reject any others. It's possible things work differently if you are applying elsewhere and I can't speak to that.