Any point in 1-3* servants?

One example that immediately comes to mind is Eurayle, who is better at dealing with male sabers then nearly any other servant in the game. That may seem niche, but it encompasses a pretty considerable portion of bosses. Another consideration is that because they're easy to get to NP5 some 3* servants can often do more damage with their noble phantasms then 4* or 5* servants, like Lu Bu, Ushiwakamaru, and Robin Hood. There's also Arash, a 1* who stands head and shoulders above everyone else in the farming utility department.

A lot of them are trash, but a good portion of them have great utility. This is coming from someone who's max ascended every 1-3* in the NA version of the game.

Here are some 1*-3* Servants I use despite having stronger ones in the same class.

David - Grants full party Evade for one turn.
Cu Chulainn - Very high survival rate. Good for long fights especially against Archer class enemies.
Medea - Rapid Words at level 4 fills her NP gauge to 100% even starting from zero, NP debuffs the enemy, and her third skill removes debuffs from one Servant.
Hans - Can buff the whole party's Crit Damage for 3 turns. Quick charging of NP with Rapid Words (75% at lvl 10) and at NP5 has a high chance of healing and buffing ATK, DEF, and Star Drop rate.

Simple, because there is no bad servant only bad master

Mashu - she is one of the best member for clearing harder content or challenges alike.
Hans - one of the best supp in the entire game.
OG Chu - one of the toughest cockroach and one of the best Lancer.
Georgios - a Tank made for various situations, he's one of the best tank along with Mashu.
Caesar - have you see his kit ? There's only damage steroid there.
Robin Hood - he's a Nuclear Bomb that can be deployed multiple times.
Blackbeard - Solid AOE Wave cleaner, basically a low rarity Drake.

Their only drawback is their stat, in terms of performance they're better or equal to most 5 Star servants.

(sorry for my english :3)
In general: there are strong point in some of them.
Basically if you lack a strong reliable nucker in some of the classes, there is:
Sabers: Caesar, Bedivere. Archers: Robin, Euryale, David. Lancers: both Cu's. Raiders: Ushiwakamaru. Assasins: Hundred Personas, Jing Ke. Casters: Medea. Berserkes: Lu Bu.
The best thing of those servants - their NP level can be raised to 5 without any quartz and, because of that, their NP Damage is higher than most of 4* and 5* NP1 servants. Sadly, this is pretty much everything this particular servants can offer - without high NP damage they loose in almost everything to most gold analogs.
But there is a pretty good bronze and silver servants that has unique niche and can be a huge help in some cases:
Arash, for example, is 1* Archer that not only outdamage most gold servants NPs, but also leave the battlefield after releasing NP that can be HUGE help in farming event quests in 3 turns (also his 3rd skill charges him 20%-30% NP gauge).
Georgios skill can force all enemies to attack only him for 3 turns and also has a heal, guts and def buff in his toolbox, so basically he is a tank. Also if you have Siegfried (that deals much more damage to dragons), Georgios NP adds a trait "Dragon" to any enemy it's been used to.
Shakespeare is a bronze Merlin with 20-40% buster buff and 20% NP charge to any ally. Medea has a skill, that charges her NP gauge from 0% to 101% on level 4 and cleanse all buffs from enemy. Hans is the best friend of crit-reliable characters with his support abilities. Cu Chulainn is a cockroach that can be hardly killed. And so on...
You can find many of them quite usable even with 5* in any class, so better have each 1*-3* servant with NP5 in your collection for some cases (not to mention that on most events bringing some of bronze and silver characters will give you event bonuses).

If you do have better servants, you should prio them for sure. Many events like christmas have a 5 star ce. That is already 60 cost gone from your max of 111 leaving only 51. That is 3 5 stars and a 1 star. Or a 5 star 2 4 stars a 3 star and a 2 star. Of course, mash is free which is why she is op.

Pretty much summed up, a bit disappointed nobody mentioned Asterios. He may be a 1 star berseker and I have him at 9/9/9. Void dust refunded thanks to babylonia main quests. He's the perfect stall team zerk. Finished Babylonia today with him in the front line coupled with Mash and sup Waver. Gorgon and Tiamat were back to back debuffed. It's a stall team though so you finish around 20+ turns. And Asterios works best if the boss or challenge quest you're doing does AoE normal attacks

If you have better options for every class, no. Heck, comparing 4* to 5* is kind of a stretch already unless you talk of extremely exceptional cases, imagine lower.

If anything Euryale is pretty good, despite she's no match for Orion. The same applies to Bedivere another exceptional 3* case.

Another thing is that 3-turn taunters are rare. There's no 5* taunter, and the only 4* is D'eon.

If you're only speaking about damage, then yes, there's no point in raising 1-3* servants since you already have different servant that can do the job better. But if you're actually looking into the utilities they bring into the team, then no, 1-3* servants are always important. Each and every servant is unique.

Just a few example:
1. Arash is the god of farming, not raising him is a dumb idea no matter what servants you have.
2. Even if you have Waver, Hans is still a very valuable asset. Unlike Waver, Hans can give you crit star through his skill and heal through his NP.
3. Leonidas is an amazing tank. Not only he taunts, he also offers crit star through his NP as well as a buster buff from his skill.
4. Matahari offers you 3 decrease defense debuff into the enemy (2 from skills, 1 from NP)
5. Mozart is able to give you a freaking amount of 50 stars just from pressing one of his skill. That's just plain amazing.

Now see from one of those examples. You can say that Tamamo is better than Mozart. But then look again, they offer a completely different things. Tamamo heals you and cut your cd, while Mozart allows you to guarantee a crit hit for 1 turn.

I'll say it again, if you're only looking for damage, then there's not really a need to raise the lower rarities one that do the same job. But if you actually want to strategize your team, then lower rarities servants are something that you must not forget as each of them bring something different from the others.

Euryale, FSN Cu and Hans are worthy of maxing skills and level.

300% OC effect of Euryale+Whim of the Goddess delivers on saber/berserker males a beyond 5* NP1/4* NP5 levels of absurd damage, plus charming and attack down.

Cu with all skills maxed will almost never dies. Perecft backliner.

Hans is a fragile crit support, with the best support skills sans Waver/Merlim/Skadi/Tamamo quartet.

Take a look on David second skill, also his NP has 2 good secondary effects (skill seal and sure hit). His caster deck can easy the arts chain of the team.

Even among the weaker ones there is good niches, Mata-Hari for 1 turn can inflict 60% def down (need ST quest and 2 of her skills maxed, plus NP fired on the same turn). Combine this with Asterios post-interlewd NP, and you can 100% def down, which is better than buff stacking some1with Waver...

Events are a big reason. Many low rarity servants have bonuses in the various events. And don't forget cost, when 5 5* CE's are needed you can't afford to bring 4 and 5* servants in your entire roster. So a selection of usable 3* at least makes events easier.