In solo queue it all depends on the flow of the game. If your team is in a defensive stance you have two choices:
- Split push hard, forcing the enemy team to either give up their numbers advantage or give up objectives.
- Join your team.
If your team is successfully defending either option is good. If you push you'll give your team a better shot at winning an engagement, and if you join your team you might tip the balance, forcing a winning teamfight and allowing your team to take more or larger objectives.
If your team is not successfully defending it's a bit different. You have to make a judgement call over whether your presence would change anything. If you think you can assist your team and keep the enemy from taking your objectives it's clearly best to go help, however if you can't do anything but can sneak one or two enemy objectives that's clearly the better option.
If your team is in an offensive stance you have to make another judgement call. If your team is a single won team fight away from victory you should group, if not you should split push and force down more objectives.
This is where things typically go wrong. It's solo queue, you're not with a team that is necessarily reliable. You have to match your team's movements, not the other way around. Split pushing requires the enemy team to be pressured elsewhere. If they are on the offensive they are pressuring themselves, but if they are on the defensive you cannot be out pushing towers when your team is not doing the same.
Good split pushing champions have either fantastic dueling potential or fantastic escapes, but you shouldn't rely on either of these. Let your team draw the enemies attention away from whatever lane you plan on pushing, and most importantly of all never assume your team will apply pressure just because you are split pushing!
Now, to respond to the OP, different champions have different playstyles inherent to their kits. A team's strengths and weaknesses are based around their champion kits, which makes each combination of champions different. A player using a split pusher needs to recognize their team's needs just as much as the rest of the team needs to recognize the strengths and weaknesses of the split pusher.
This is an issue that should be solved in champion select or not at all. If you do not want to face a 4v5 for any reason (which is silly and reduces the amount of flexibility your team has strategy wise) make this known before someone picks a strong split pusher. If someone picks a split pusher anyway suck it up, it's not your job to impose your opinions on someone else. If they are a good split pusher they will do their job while still assisting the team, and if not they do not fully understand the role they have chosen, and that's a detriment dealt out by matchmaking alone.
TLDR: Split pushers should inform their team in champion select so they can pick a strong 4v5 comp. Players uncomfortable in handling 4v5s should inform their team so split pushers can be avoided. You only get to play your role, so if your opinion gets overruled just let it go and try your best. In the end there are always bad players who don't do what their team needs, however no strategy should be judged based on the actions of these bad players.