In our 5e Curse of Strahd campaign, I awarded XP and kept everyone the same level. One of the players (of 4) had to miss quite a few sessions because of a variable works schedule that took him out of town. His character would either tag along as an NPC that the other players could run in combat or disappear for a solo quest of hunting wolves. Worked out just fine.
For other 5e campaigns, I've awarded XP for the characters who participated in the session. Other characters simply fade into the background for whatever reason. We don't think about it too hard. All the players are on board with that and they don't consider it a punishment to miss out on XP. It's seen as a reward to the character for participating in the adventure that session. In one of my current campaigns, the gap has been a large as 4 levels. As the levels get higher and it takes more XP to advance, the gap has been shrinking. They are now all between 11th and 13th. Everyone has missed a game now and then, some more than others, but the party marches on. Has been working out just fine.
There hasn't been a problem with power disparity in our 5e games due to characters being different levels - I frankly think that this may be some holdover thinking from prior editions where the level discrepency was more pronounced. Not every group of monsters is the same exact level (e.g. hobgoblin captain, 2 hobgoblins, and 3 wolves), so I don't see the necessity of the PCs being the same level either. Sometimes they have a low level NPC along for the ride fighting with them or that they need to protect. Again, not really an issue as long as everyone can contribute to the game session in a meaningful way - something I make sure of as a DM by spreading the spotlight around. I don't try too hard to balance the encounters either. I certainly want to make sure the party is sufficiently challenged most of the time, but I also throw some intentionally easy encounters at them - and excessively difficult ones - once in a while to keep them guessing. Point being, their exact PC levels don't matter too much in my encounter design. There are plenty of ways to ratchet up or down the difficulty on the fly, if I feel a particular scene warrants it.
Finally, I should mention that I award XP not just for killing monsters. Disabling/avoiding traps, solving puzzles, making allies, talking themselves out of trouble, discovering plot points, completing mini-quests, etc etc are all rewarded with XP to some extent. I hand out the XP after each session to the PCs who participated and the players track it. Simple. They all get excited when they are within reach of leveling up - and some happily announce it at the start of a session. Seems to work well for our group.