It's very unlikely for any male to hit a C5 like that and have a range going to as low as G2 without being in pure falsetto or head voice at C5.
Especially Jon.
There's a thing called passagio which is around anything from the C4 - E4/F4 area for males and second passagio is slightly higher and most slightly trained singers can go up to second passagio without flipping to falsetto.
Once you do some vocal exercises to get rid of that flip (Second passagio) you can go into your highest parts of mixed voice which can sound exactly like chest voice.
The reason Adam Levine, etc mix voice sound very thin like that is due to a high larynx position (Most modern pop singers usually sing with high larynx position).
The reason Jon's mix sounds chesty like what it was or is is due to only a slightly higher larynx than normal, in around 2007 (Maybe even slightly before) he trained it to become at a neutral or even low position which made him sound kinda operatic at even his high notes.
5:55 until the end is where Jon does those "ooooh" stuff at the end of TAALS and slides from anything from D4 to G#4 and once or twice slightly higher than A4.
Your able to do that once you have mix as a man especially if you have a baritone lower range (Jon). If you didn't have mix voice you would likely have to shout it too loud or even scream it to sound chesty there unless you were a woman.
Many tenors aren't able to go above E4 - F4 and those areas without mixed voice.
Also if you didn't know.. It sometimes happens that tenors think they are baritones and actually have voices that behave like that but once they get vocal training, etc they find out out they are a tenor and can sing tenor range due to development of the voice.
If Jon actually shouted and screamed everything his voice would have been gone already at the first album or second one.
Jon does pull up chest to the C4 - F4 area though which is another thing, still isn't pure chest, has some mix in it though.
Before I trained for mixed voice in real life I could only go to E4 - F4 by shouting loudest I can, now I can sing at C4 - G#4 or even A4 and sound very clean (1995 clean like Jon) easy and still sound chesty, which is really tenor singing range.
Now the problem is Jon really has forgotten about mix voice or just uses it to a minimum amount unless he needs to. Which leads to him shouting everything above G4.
that's for example why he shouts out "Bed of nails" nowadays or really shouts "Lay my hands on youuuuuuu" after the solo, and many more examples I could give. And the thing of avoiding mix voice for highest notes is a recipe for vocal damage.
Jon was techincally most correct around 2008 LH era.
2013 and afterwards he really ignored technique and stopped using the twang he used.
The Jon in 2018 we see nowadays is absolutely the same I did before I developed my mix voice the most I could.. I had to shout G#4 note which Jon does in LYHOM after solo ("Youuuuu" after "Lay my hands on").
My only current problem with mix voice right now is a too high larynx which leads to me sounding very thin above A4, but when I have a lower larynx than usual I sound chestier than normal above A4.
My talking range, normal range, etc is pretty much almost the same as what Jon had 95/96 or maybe 97/98. Have to sing "Only the names will change" the same way Jon did to reach the F#4, etc without being too loud.
Jon has extremely likely gotten vocal training before Bon Jovi the band formed.
When people say that tenors can hit C5, that's really with training and the C5 must sound good enough to be counted as a part of your singing range. Many completely untrained baritones or tenor can't really go above E4 - F4 area unless they are shouting too loud or maybe even screaming.
For your "Straining and belting to the absolute max":
You can use mix voice almost like how you can use chest voice, you can scream it like a drunk angry man, you can abuse it and cause vocal irritation/damage. The difficulties when finding your mix voice the first time could be controlling the notes you hit or adding rasp without screaming, etc to it. For example in opera you usually train your mixed voice to sound as chesty as your chest voice as possible, which leads to many people thinking that they are hitting that D5 in actual chest voice.
Mixed voice is actually also kind of proportional to your chest voice, the lower your chest voice the lower your mixed voice usually is.
Inevitably he will have to lower notes at some point though due to the chest/mix voice thing being kinda proportional.
If you are too quiet with mix voice you will only most of the times really only hear head voice/falsetto, being loud with it in the same way as Jon leads to it being chesty sounding. Accessing mixed voice isn't really the same as falsetto or head voice though.
Jon going to his higher part of mix voice (First 2 albums are very good examples and sometimes Power Station):
2:20 - 2:23 going to his higher part of mix voice.
Sorry for the wall of text, only wrote the interesting stuff though, avoided everything boring. Hard to explain singing stuff without writing too much, due to how singing has misconceptions, truth's and a lot of other stuff.