Why do legislatures give a tiny subset of their members veto power over a bill? Isn't that undemocratic? Why can't they simply debate and vote on every bill introduced as a whole?
Committees exist primarily to increase the efficiency with which legislative bodies can review potential legislation. Carefully considering a bill is a time consuming process, and the more people there are who are involved in it, the longer it takes. A legislature with a dozen committee can review more than twelve times as many bills as a legislature without committees, and if the committees are specialized by subject-matter, as they usually are, the initial review of the bill is from members with the most subject matter expertise in the matters addressed by the bill. Generally speaking, it would be impossible for the legislature as a whole to consider every bill introduced by its members in any reasonable level of depth.
Committees prioritize the scarce available time for the legislature as a whole to bills that have a reasonable chance of being adopted and that have already had any glitches in their details worked out, so that the legislature as a whole is primarily considering the overall policies behind a bill that has substantial support, rather than getting bogged down considering bills that don't have serious support or that have many minor drafting problems.
Committees also make it easier to identify when there are multiple bills on the same or similar subject-matter that can be be combined in some fashion, or would lead to contradictions in policy if all of them were adopted.
Also, committee membership is invariably structured so that the partisan divide in each committee parallels that of the body as a whole, and so that the members appointed to a committee by the leadership of each party are representative of the views of the party as a whole on the subject-matter of the kind of bills considered by that committee (and the committee members also tend to be people that their peers would have deferred to for guidance on those kinds of bills). So, it is quite unlikely that a bill that would have passed in the legislative body as a whole would be defeated in committee. This greatly reduces the extent to which a committee, which could be undemocratic in theory, will, in practice, act in a manner that does not reflect the wishes of the majority.
Moreover, most legislatures have a process by which a bill which a committee is refusing to address for some reason can be pulled out of committee for consideration by the legislative body as a whole if it wishes to do so.
Why do legislatures give a tiny subset of their members veto power over a bill?
As a general rule, they don't. The legislative leadership generally has the power to bypass committees if they want to do so. If a committee is simply never considering a bill, then chances are that the leadership of that legislature simply doesn't want to consider the bill. They may prefer to have it die in committee where there is less attention than to vote it down as a whole, but that's their preference, not a requirement.
This also allows individual legislators to simply say that it is out of their control.
tl;dr - Conducting all business on the floor is impractical and inefficient. Committees allow legislators to make better usage of their time and skills.
Working Bills is Hard with Large Groups
Legislative committees don't just vote on bills. One of their major functions is as a quality assurance tool. In committee, legislators actually edit the text of bills. This is commonly called "working the bill" though it may have a different name in different legislatures.
This is one important reason committees exist: trying to work a bill on the floor with 100+ people in attendance would be very difficult.
Being a legislator is difficult because you will be called on to vote on significant public issues which you don't know much about. There really is no avoiding this problem because no single person can be expected to have an in-depth understanding of all the issues a society deals with. Committees allow legislators to develop some expertise on a single set of issues.
This effect is even more important if your legislature has term limits. Legislators take time to learn how the legislature works and how their policy environment works. Committees help them learn it.
Time and Opportunity
ohwilleke's answer already hit on this, but I would like to expand it a bit. Committees don't just spend time working bills and voting, they may also spend significant amounts of time hearing testimony on issues related to their bills. Time on the floor is incredibly valuable, so without committees there would be less opportunity for debate, discussion, and testimony.
In the Indian Parliamentary System, it is more or less same as Westminister type, and parliamentary committees exists for the following reasons:
1. Expert Presence - usually, the elected people are not technocrats, so if parliament needs technocrats for consultation on the wide fields like AI, Space, 5G then they can call subject experts and get technical nuances in framing the legislation.
2. Consensus reaching mechanism - The legislation presented before parliament can go for detailed scrutiny before the select committee or joint parliamentary committee, which will do a detailed analysis of the bill and come up with consensus between different political parties.
3. Promote internal Democracy in Parties - rules like the anti-defection law doesn't kick in as there is no whip for the committee discussion and members of the committee can speak freely.
4. Reduces Parliament workload - The parliament setting is usually 100 days in a year. So if each bill was framed during the normal plenary session of parliament, there would not be enough time for import business like budget presentations, votes on account, demands for grant, etc.
Legislative committees have several advantages over plenary sessions when dealing with fine-tuning legislation: first, they tend to be smaller; and second, they can work under different rules than their parent body.
Regarding the size of the committee, as I mentioned it is generally smaller but every legislative body that I know of provides for a committee of the whole which can run under committee rules, which I'll discuss next. They aren't used very often, but I mention this to show that committees can be any size that a legislative body wishes.
Committees also enjoy the ability to have more leeway regarding how they dispose of matters. For example, calling witnesses tends to be difficult in a plenary session (mostly because the rooms aren't really designed for it), but committees handle this easily. These committees also may have staff who can advise on difficult or complicated matters, which the plenary would not have easy access to. Also, committees can travel, something that would be very difficult for a plenary body to do.
Lastly, committees can't veto anything: all that they can do is present a report back to the plenary indicating whether they recommended something or not, and give reasoning and potential amendments. It's up to the plenary to agree with them or not.